# An algorithm for constructing graphs with given eigenvalues and angles

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 An algorithm for constructing graphs with given eigenvalues and angles Dragan Stevanović Department of Mathematics, Faculty of Philosophy, Ćirila i Metodija 2, Niš, Serbia, Yugoslavia Abstract Let the eigenvalues of a graph and the angles between eigenspaces and the co-ordinate axes of the corresponding real vector space be given for a graph. Cvetković [2] gave a method of constructing a graph which is the supergraph of all graphs with given eigenvalues and angles. Based on this, we describe a branch & bound algorithm for constructing all graphs with given eigenvalues and angles. 1 Introduction Let G be the graph on n vertices with adjacency matrix A. Let {e 1, e 2,..., e n } constitute the standard orthonormal basis for R n. Then A has spectral decomposition A = µ 1 P 1 + µ 2 P µ m P m, where µ 1 > µ 2 >... > µ m and P i represents the orthogonal projection of R n onto E(µ i ) (moreover Pi 2 = P i = Pi T, i = 1,..., m; and P i P j = O, i j). The nonnegative quantities α ij = cos β ij, where β ij is the angle between E(µ i ) and e j, are called angles of G. Since P i represents the orthogonal projection of R n onto E(µ i ) we have α ij = P i e j. The sequence α ij (j = 1, 2,..., n) is the ith eigenvalue angle sequence; α ij (i = 1, 2,..., m) is the jth vertex angle sequence. Cvetković ([1]) gave the first algorithm for construction of trees with given eigenvalues and angles. Then in [2] he gave a method that uses only eigenvalues and angles to construct the graph which is a supergraph of all graphs with given eigenvalues and angles. Such a supergraph is the quasi-graph in general case, which is described in Section 5. If we also know the eigenvalues and angles of the complementary graph, we can construct the fuzzy image of a graph, which enhances the quasi-graph. In the case of trees, that supergraph is the quasibridge graph, whose construction is much simpler than that of quasi-graph and fuzzy image. It is described in Section 4. 1

2 Further, in [3] Cvetković gave a lower bound on the distance between vertices based on eigenvalues and angles of graph. In Section 3 we give a new lower bound, similar to this one and show that the two are independent of each other. We also present some statistical data on random graphs. Based on the lower bound on distance and the supergraph of all graphs with given eigenvalues and angles, in Section 6 we give a branch & bound algorithm to construct all graphs with given eigenvalues and angles. 2 Preliminary Lemmas If a graph or vertex invariant can be determined provided the eigenvalues and angles are known, then the invariant is called EA-reconstructible ([1]). The basic property of angles is given in the next lemma. Lemma 2.1 ([7]) The number of closed walks of length s starting and terminating at vertex j is given by m i=1 µs i α2 ij. Corollary 2.2 ([7]) The degree d j of the vertex j, and the number t j of triangles containing the vertex j, are given by d j = m αijµ 2 2 i, t j = 1 2 i=1 m αijµ 2 3 i. A partition of the vertex set of G is called admissible if no edge of G connects vertices from different parts; and subgraphs induced by the parts of an admissible partition are called partial graphs (thus a partial graph is a union of components, and the components are induced by the parts of the finest admissible partition). The spectra and angles of these partial graphs are called the partial spectra and partial angles corresponding to the original partition. Lemma 2.3 ([6]) Given the eigenvalues, angles and an admissible partition of the graph G, the corresponding partial spectra and partial angles of G are determined uniquely. Theorem 2.4 ([6]) Given the eigenvalues and angles of a graph G, there is a uniquely determined admissible partition of G such that (i) in each partial graph all components have the same index, and i=1 (ii) any two partial graphs have different indices. For further properties of angles, see the monograph [7]. 2

3 w u v Figure 1: The tree from example. 3 Lower bounds on distance From the spectral decomposition of A we have a (s) jk P i e j P i e k P i e j P i e k we get a (s) distance between vertices j and k in G. jk = m i=1 µs i P ie j P i e k. Since m i=1 µs i α ij α ik. Let d(j, k) be the Lemma 3.1 ([3]) If g = min {s : m i=1 µs i α ij α ik 1}, then d(j, k) g. Lemma 3.2 If g = min {s : m i=1 µs+2 i α ij α ik d j + d k + δ s1 s}, where δ s1 is the sum of s 1 smallest degrees of vertices other than j and k, then d(j, k) g. Proof Let j = w 0, w 1,..., w d(j,k) = k be the shortest path between j and k. The number of paths of the length d(j, k) + 2 between j and k which have the form j = w 0,..., w i, u, w i,..., w d(j,k) = k, where u is arbitrary neighbor of w i, is at least d j + d k + δ d(j,k)1 d(j, k). Hence, d(j, k) g. Example. For the tree shown in Fig. 1, from Lemma 3.1 we have d(u, v) 2, while from Lemma 3.2 it follows that d(u, v) 3. On the other hand, from Lemma 3.1 it follows that d(w, v) 3, while Lemma 3.2 gives only that d(w, v) 1. This shows that the lower bounds given in these lemmas are independent of each other. In order to get a better lower bound, one must then take the greater of the values given by these lemmas. The average value of the lower bound (obtained from lemmas 3.1 and 3.2) in connected random graphs having from 10 to 40 vertices and given number of edges (n 1, n, 2n, 3n, n log n, n n i n 2 /4) is shown in Table 1. For each number of edges, 100 connected random graphs were taken. From this table it can be seen that the average value of the lower bound increases with the number of vertices in graphs having O(n) edges, while it decreases in graphs with at least O(n log n) edges. The exact boundary where this average value is constant is somewhere between O(n) and O(n log n). The average value of the lower bound for all pairs of vertices at given distance in connected random graphs with 30 vertices and given number of edges is shown in Table 2. As before, for each number of edges, 100 connected random graphs 3

4 were taken. The symbol in d th row means that in the set of random graphs there was none with diameter d. From this table we can see that in all cases the ratio between the lower bound and the distance decreases with the increase of distance. It also shows that the lower bound is almost unusable in graphs with at least O(n n) edges. vertices n 1 n 2n 3n n log n n n n 2 /4 10 2,06 1,94 1,19 1,03 1,11 1,02 1, ,33 2,26 1,32 1,07 1,11 1,01 1, ,50 2,40 1,41 1,10 1,10 1,01 1, ,67 2,56 1,46 1,12 1,09 1,00 1, ,79 2,68 1,51 1,13 1,07 1,00 1, ,87 2,81 1,55 1,14 1,06 1,00 1, ,99 2,90 1,58 1,16 1,06 1,00 1,00 Table 1: Average value of lower bound on distance. distance n 1 n 2n 3n n log n n n n 2 /4 1 1,00 1,00 1,00 1,00 1,00 1,00 1,00 2 1,78 1,76 1,31 1,11 1,06 1,00 1,00 3 2,24 2,20 1,57 1,31 1,21 1,01 4 2,61 2,55 1,98 1,89 1,95 5 2,95 2,88 2,55 2,33 6 3,23 3,16 3,13 7 3,47 3,42 4,00 8 3,64 3,65 9 3,79 3, ,91 4, ,76 4, ,41 4, ,79 4, ,00 15 Table 2: Average value of lower bound for vertices at given distance. 4 Quasi-bridge graphs The following theorem is proved in [2]. 4

5 Theorem 4.1 Let uv be a bridge of a graph G. Then P 2 G + 4P GuP Gv is a square. The necessary condition for two vertices u and v to be joined by a bridge, provided by this theorem, is called the bridge condition. The quasi-bridge graph QB(G) of the graph G is defined as the graph with the same vertices as G, with two vertices adjacent if and only if they fulfil the bridge condition. Define a quasi-bridge as an edge of QB(G). If G is a tree, then we obviously have that G is a spanning tree of QB(G). The bridge condition is not sufficient for the existence of the bridge. On the other hand, there are trees for which the equality QB(G) = G holds. Such examples are the stars S n and the double stars DS m,n with m n (see [9]). Here we deal with the number of quasi-bridges in trees. In Table 3 we give the statistical results obtained by determining the number of quasi-bridges in random trees which had from 5 to 30 vertices, where for each number of vertices we randomly chose 100 trees. The number of vertices is shown in the first column, while the minimum, maximum and the average number of quasi-bridges in these trees are shown in the second, third and fourth column, respectively. Data from this table show that many small random trees satisfy QB(G)=G. On the other hand, Cvetković [1] showed that for almost every tree G there is a nonisomorphic cospectral mate G with the same angles. Hence, both G and G must be spanning trees of QB(G ), and for almost every tree QB(G) G. vertices min max average vertices min max average Table 3: Quasi-bridges in random trees. Although it cannot be seen from Table 3 it is the case that e(qb(t )) = Θ(e(T ) 2 ), where e(g) is the number of edges of G. One example is shown in Fig. 2. It is a rooted tree of depth 2 where the root has a descendants, and 5

6 each of them has exactly one descendant. All neighbors of the root are similar, hence have the same vertex-deleted characteristic polynomial. The same holds for all leaves. Since the bridge condition is satisfied for at least one pair of vertices consisting of a neighbor of a root and a leaf, it is also satisfied for all pairs of vertices consisting of an arbitrary neighbor of a root and an arbitrary leaf. Hence the tree T from Fig. 2 is a spanning subgraph of QB(T ). Further examples consist of rooted trees regular in the following sense: all nodes on the same level have the same number of descendants. T e(t ) = 2a T... QB(T )... e(t ) = a 2 + a Figure 2: QB(T ) can have many edges. 5 Quasi-graphs and fuzzy images The following theorem is proved in [2]. Theorem 5.1 Let G be a graph with n vertices and m edges, and let uv be an edge of G. Then there exists a polynomial q(x) of degree at most n 3 such that (x n (m 1)x n2 + q(x))p G (x) + P Gu (x)p Gv (x) is a square. The necessary condition for two vertices u and v to be adjacent, provided by this theorem, is called the edge condition. The quasi-graph Q(G) of the graph G is defined as the graph with the same vertices as G, with two vertices adjacent if and only if they fulfil the edge condition. Obviously, any graph is spanning subgraph of its quasi-graph. If G is regular and both G and G are connected then from the eigenvalues and angles of G we also know the eigenvalues and angles of G [6]. Now, the edge condition in G is a necessary condition for non-adjacency in G, and any two distinct vertices of G are adjacent either in Q(G) or in Q(G). If they are adjacent in one and not adjacent in the other, then their status coincides with that in Q(G). Thus, the fuzzy image F I(G) is defined as the graph with the same vertex set as G and two kinds of edges, solid and fuzzy. Vertices u and v of F I(G) are joined by a fuzzy edge if they are adjacent in both Q(G) and Q(G), 6

7 otherwise they are joined by a solid edge if they are adjacent in Q(G) and they are non-adjacent if they are non-adjacent in Q(G). Except for small values of n (up to 4), it is very difficult to use the edge condition practically. Since the coefficients of the characteristic polynomials are integers, we can use the following weaker corollary. Corollary 5.2 Let G be a graph with n vertices, and let uv be an edge of G. Then P Gu (n)p Gv (n) is quadratic residue modulo P G (n) for every n Z. Using Corollary 5.2 involves a loss of information. Indeed, for regular graphs we got that usually only a few pairs of vertices are not joined by a fuzzy edge in the fuzzy image. Thus the problem of implementing the edge condition still remains. 6 The Constructing Algorithm The algorithm presented at the end of this section is of branch & bound type. It does not assume anything about graph connectedness, but in the case of nonconnected graphs, we can simplify the construction. Namely, we can find the partial eigenvalues and angles from Theorem 2.4. Then we construct all partial graphs with the corresponding eigenvalues and angles, after which we have to construct all the combinations of the partial graphs obtained. Before the algorithm enters the main loop, it determines the graph G that is the supergraph of all graphs with the given eigenvalues and angles. Let G denote the putative graph with given eigenvalues and angles, with vertex set V (G) and edge set E(G). In the case of trees we have G = QB(G), in the case of regular connected graphs with connected complements we have G = F I(G), while in other cases G = Q(G). Then for each pair (u, v) of vertices, the algorithm determines the lower bound d l (u, v) on the distance in G between vertices u and v, based on Lemmas 3.1 and 3.2. At level 0 we choose the vertex v 1 arbitrarily, and pass to level 1, where we enter the main loop. When we come to level i (i 1) we have already constructed the subgraph G i induced by vertices v 1, v 2,..., v i. Then we choose the vertex v i+1 from the set of remaining vertices, select its neighbors from the set {v 1, v 2,..., v i }, and pass to the next level, until G is constructed in whole. The fact that at each level we know the induced subgraph of G provides us with the possibility of using the following well-known theorem. Theorem 6.1 (see, for example, [8], p. 119) Let A be a Hermitian matrix with eigenvalues λ 1 λ 2... λ n and let B be one of its principal submatrices. If the eigenvalues of B are ν 1 ν 2... ν m then λ i ν i λ nm+i (i = 1, 2,..., m). The inequalities of Theorem 6.1 are known as Cauchy s inequalities and the whole theorem as the Interlacing theorem. If Cauchy s inequalities are not 7

9 If we select the vertex v as v i+1 then the number of neighborhoods of v in the set {v 1, v 2,..., v i } that have to be examined at this level is equal to N v = = = ( f v m v ( f v m v ( f v m v ) ( f + v m v + 1 ) ( f + v m v ) ) f v m v m v + 1 )( 1+ f v m v m v +1 ( f v M v ) ( f v m v ( f v M v +2 M v 1 ) (f v m v )... (f v M v +1) (m v +1)... M v ( 1+ f v M v +1 M v )... )). Hence, as the vertex v i+1 we choose the vertex v which minimizes the value N v (in the algorithm we use the value log N v computed using Stirling s formula). Once the vertex v i+1 is selected, we have to specify its neighbors from the set {v 1,..., v i } in order to construct the graph G i+1 completely. Let us introduce the following conditions that are applied in the algorithm: (i) the degree condition at level i is satisfied if for every vertex v of G i+1 the degree of v in G i+1 is not greater than the degree of v in G (see Corollary 2.2). (ii) the triangle condition at level i is satisfied if for vertex v of G i+1 the number of triangles in G i+1 containing v is not greater than the number of triangles of G containing v (see Corollary 2.2). (iii) the distance-3 condition at level i is satisfied if for every pair (u, v) of vertices of G i+1 for which d l (u, v) 3 holds, u and v do not have a common neighbor in G i+1. Denote by F i+1 the set of forced vertices from {v 1,..., v i } that are neighbors of v i+1 in G. In case that G = F I(G) we also put into F i+1 those vertices from {v 1,..., v i } that are connected by a solid edge to v i+1 in F I(G). Of course, vertices from F i+1 must be adjacent to v i+1 in G i+1. If any of the above conditions is not satisfied when we join v i+1 to the vertices from F i+1 then we have to return to the previous level. The neighborhood S {v 1,..., v i } of v i+1 consisting of nonforced vertices must be such that also none of above conditions is broken. If any of the conditions is broken, we have to select the lexicographically nearest neighborhood satisfying them. This new neighborhood can not be a superset of one that does not satisfy them, due to the monotonicity of the conditions. If there is no such neighborhood, we have to return to the previous level. We return to the previous level in the backward phase of the algorithm. Notice that when we are looking for the next neighborhood of nonforced vertices to be examined we allow it to be the superset of the current one. 9

10 The Constructing Algorithm Input: Eigenvalues and angles of a graph. Output: All graphs with given eigenvalues and angles. begin find the supergraph G for all pairs (u, v) find d l (u, v) choose vertex v 1 i=1 1. Forward phase while i > 0 if i is equal to the number of vertices then if the constructed graph has given eigenvalues and angles then print the graph else check whether any of the vertices v 1,..., v i is forced at this level for each v V (G) V (G i ) find the smallest m v and the largest M v possible for the degree of v in G i+1 if Cauchy s inequalities hold for G i and m v M v for each v V (G) V (G i ) then select vertex v i+1 determine the set F i+1 if F i+1 does not break any condition then find the lexicographically smallest neighborhood S i+1 of v i+1 consisting of non-forced vertices of G i (m vi+1 S M vi+1 ) while S i+1 does not satisfy the conditions find the next such neighborhood S i+1 of v i+1 that is not a superset of the previous one if S i+1 exists then G i+1 G i {v i+1 } (F i+1 S i+1 ) i i + 1 goto 1. i i 1 2. Backward phase while i > 1 if some vertex became forced at level i + 1 then it is no longer forced find the next set S i+1 of non-forced neighbors that may be a superset of the previous one while S i+1 does not satisfy the conditions 10

11 end. find the next set S i+1 of non-forced neighbors that is not a superset of the previous one if S i+1 exists then G i+1 G i {v i+1 } (F i+1 S i+1 ) i i + 1 goto 1. else i i 1 goto 2. We implemented this algorithm and it may be obtained from In [5] it is found that there are 58 pairs of cospectral graphs with the same angles on 10 vertices. However, the graphs from some 29 pairs are the complements of graphs from other 29 pairs. We tested the algorithm on those pairs of graphs which edge density is less than 1/2. For testing we used Pentium MMX machine on 200 MHz. The average of running times in seconds for both graphs in each pair is given in Table 4. From this table we can see that the running time depends mostly on the difference between the number of edges in the graph and the supergraph G. To speed up calculations, the algorithm checks the fulfillness of Cauchy s inequalities after constructing the induced subgraph on [2n/3] vertices. This checking is very time-consuming and it is the best to use it only on one level during the construction, but not too early since it will rarely happen that the Cauchy s inequalities are not fulfilled when less than half of the graph is constructed. The choice of [2n/3] vertices for these graphs gave the improvement in running time ranging from 5% to 2000% compared to the case when the Cauchy s inequalities are not checked. The distance-3 condition was of no help in this case since there are no pairs of vertices in these graphs for which the lower bound on distance is at least 3. However, the lower bound on distance helps to remove many edges from the supergraph G. The hardest pair of graphs is no. 11, which is the only pair of regular graphs. All pairs of their vertices are joined by fuzzy edge in the fuzzy image and the lower bound on distance is trivial, so that the algorithm has to check all the graphs on 10 vertices with given vertex degrees and the numbers of triangles passing through them. Acknowledgements. This paper is part of author s M.Sc. thesis, written under the supervision of Prof. Dragoš Cvetković at the University of Niš. The author is greatly indebted to the anonymous referees for several helpful comments which led to the considerable improvement of paper. 11

12 pair time pair time pair time >10h References Table 4: The average running times for pairs of graphs from [1] Cvetković D., Constructing trees with given eigenvalues and angles, Linear Algebra and Appl., 105 (1988), 1 8 [2] Cvetković D., Some possibilities of constructing graphs with given eigenvalues and angles, Ars Combinatoria, 29A (1990), [3] Cvetković D., Some comments on the eigenspaces of graphs, Publ. Inst. Math.(Beograd), 50(64) (1991), [4] Cvetković D., Doob M., Some developments in the theory of graph spectra, Linear and Multilinear Algebra, 18 (1985), [5] Cvetković D., Lepović M., Cospectral graphs with the same angles and with a minimal number of vertices, Univ. Beograd. Publ. Elektrotehn. Fak., Ser. Mat. 8 (1997), [6] Cvetković D., Rowlinson P., Further properties of graph angles, Scientia (Valparaiso), 1 (1988), [7] Cvetković D., Rowlinson P., Simić S., Eigenspaces of Graphs, Encyclopedia of Mathematics and Its Appl., Vol. 66, Cambridge University Press, 1997 [8] Marcus M., Minc H., A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Inc., Boston, Mass., 1964 [9] Stevanović D., Construction of graphs with given eigenvalues and angles (Serbian), M.Sc. Thesis, University of Niš, Niš,

### 10. Graph Matrices Incidence Matrix

10 Graph Matrices Since a graph is completely determined by specifying either its adjacency structure or its incidence structure, these specifications provide far more efficient ways of representing a

### A CHARACTERIZATION OF TREE TYPE

A CHARACTERIZATION OF TREE TYPE LON H MITCHELL Abstract Let L(G) be the Laplacian matrix of a simple graph G The characteristic valuation associated with the algebraic connectivity a(g) is used in classifying

### SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

### Theorem A graph T is a tree if, and only if, every two distinct vertices of T are joined by a unique path.

Chapter 3 Trees Section 3. Fundamental Properties of Trees Suppose your city is planning to construct a rapid rail system. They want to construct the most economical system possible that will meet the

### Lecture 7: Approximation via Randomized Rounding

Lecture 7: Approximation via Randomized Rounding Often LPs return a fractional solution where the solution x, which is supposed to be in {0, } n, is in [0, ] n instead. There is a generic way of obtaining

### Solutions to Exercises 8

Discrete Mathematics Lent 2009 MA210 Solutions to Exercises 8 (1) Suppose that G is a graph in which every vertex has degree at least k, where k 1, and in which every cycle contains at least 4 vertices.

### Approximation Algorithms

Approximation Algorithms or: How I Learned to Stop Worrying and Deal with NP-Completeness Ong Jit Sheng, Jonathan (A0073924B) March, 2012 Overview Key Results (I) General techniques: Greedy algorithms

### Good luck, veel succes!

Final exam Advanced Linear Programming, May 7, 13.00-16.00 Switch off your mobile phone, PDA and any other mobile device and put it far away. No books or other reading materials are allowed. This exam

### Approximating the entropy of a 2-dimensional shift of finite type

Approximating the entropy of a -dimensional shift of finite type Tirasan Khandhawit c 4 July 006 Abstract. In this paper, we extend the method used to compute entropy of -dimensional subshift and the technique

### Introduction to Flocking {Stochastic Matrices}

Supelec EECI Graduate School in Control Introduction to Flocking {Stochastic Matrices} A. S. Morse Yale University Gif sur - Yvette May 21, 2012 CRAIG REYNOLDS - 1987 BOIDS The Lion King CRAIG REYNOLDS

### COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

### Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS

Homework MA 725 Spring, 2012 C. Huneke SELECTED ANSWERS 1.1.25 Prove that the Petersen graph has no cycle of length 7. Solution: There are 10 vertices in the Petersen graph G. Assume there is a cycle C

### Bindel, Fall 2012 Matrix Computations (CS 6210) Week 8: Friday, Oct 12

Why eigenvalues? Week 8: Friday, Oct 12 I spend a lot of time thinking about eigenvalue problems. In part, this is because I look for problems that can be solved via eigenvalues. But I might have fewer

### 4 Basics of Trees. Petr Hliněný, FI MU Brno 1 FI: MA010: Trees and Forests

4 Basics of Trees Trees, actually acyclic connected simple graphs, are among the simplest graph classes. Despite their simplicity, they still have rich structure and many useful application, such as in

### (67902) Topics in Theory and Complexity Nov 2, 2006. Lecture 7

(67902) Topics in Theory and Complexity Nov 2, 2006 Lecturer: Irit Dinur Lecture 7 Scribe: Rani Lekach 1 Lecture overview This Lecture consists of two parts In the first part we will refresh the definition

### GRAPH THEORY and APPLICATIONS. Trees

GRAPH THEORY and APPLICATIONS Trees Properties Tree: a connected graph with no cycle (acyclic) Forest: a graph with no cycle Paths are trees. Star: A tree consisting of one vertex adjacent to all the others.

### Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

### Chapter 4. Trees. 4.1 Basics

Chapter 4 Trees 4.1 Basics A tree is a connected graph with no cycles. A forest is a collection of trees. A vertex of degree one, particularly in a tree, is called a leaf. Trees arise in a variety of applications.

### On the Randić index and Diameter of Chemical Graphs

On the Rić index Diameter of Chemical Graphs Dragan Stevanović PMF, University of Niš, Niš, Serbia PINT, University of Primorska, Koper, Slovenia (Received January 4, 008) Abstract Using the AutoGraphiX

### Summary of week 8 (Lectures 22, 23 and 24)

WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry

### 1 Basic Definitions and Concepts in Graph Theory

CME 305: Discrete Mathematics and Algorithms 1 Basic Definitions and Concepts in Graph Theory A graph G(V, E) is a set V of vertices and a set E of edges. In an undirected graph, an edge is an unordered

### Planar Tree Transformation: Results and Counterexample

Planar Tree Transformation: Results and Counterexample Selim G Akl, Kamrul Islam, and Henk Meijer School of Computing, Queen s University Kingston, Ontario, Canada K7L 3N6 Abstract We consider the problem

### Labeling outerplanar graphs with maximum degree three

Labeling outerplanar graphs with maximum degree three Xiangwen Li 1 and Sanming Zhou 2 1 Department of Mathematics Huazhong Normal University, Wuhan 430079, China 2 Department of Mathematics and Statistics

### GRAPH THEORY LECTURE 4: TREES

GRAPH THEORY LECTURE 4: TREES Abstract. 3.1 presents some standard characterizations and properties of trees. 3.2 presents several different types of trees. 3.7 develops a counting method based on a bijection

### AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS

Discussiones Mathematicae Graph Theory 28 (2008 ) 345 359 AN UPPER BOUND ON THE LAPLACIAN SPECTRAL RADIUS OF THE SIGNED GRAPHS Hong-Hai Li College of Mathematic and Information Science Jiangxi Normal University

### Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

### Inner Product Spaces

Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

### Lesson 3. Algebraic graph theory. Sergio Barbarossa. Rome - February 2010

Lesson 3 Algebraic graph theory Sergio Barbarossa Basic notions Definition: A directed graph (or digraph) composed by a set of vertices and a set of edges We adopt the convention that the information flows

### GRADUATE STUDENT NUMBER THEORY SEMINAR

GRADUATE STUDENT NUMBER THEORY SEMINAR MICHELLE DELCOURT Abstract. This talk is based primarily on the paper Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees by Marcus, Spielman, and Srivastava

### 1 Singular Value Decomposition (SVD)

Contents 1 Singular Value Decomposition (SVD) 2 1.1 Singular Vectors................................. 3 1.2 Singular Value Decomposition (SVD)..................... 7 1.3 Best Rank k Approximations.........................

### Similarity and Diagonalization. Similar Matrices

MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Trees and Fundamental Circuits

Trees and Fundamental Circuits Tree A connected graph without any circuits. o must have at least one vertex. o definition implies that it must be a simple graph. o only finite trees are being considered

### Numerical Analysis Lecture Notes

Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number

### SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS. Nickolay Khadzhiivanov, Nedyalko Nenov

Serdica Math. J. 30 (2004), 95 102 SEQUENCES OF MAXIMAL DEGREE VERTICES IN GRAPHS Nickolay Khadzhiivanov, Nedyalko Nenov Communicated by V. Drensky Abstract. Let Γ(M) where M V (G) be the set of all vertices

### 2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

### IE 680 Special Topics in Production Systems: Networks, Routing and Logistics*

IE 680 Special Topics in Production Systems: Networks, Routing and Logistics* Rakesh Nagi Department of Industrial Engineering University at Buffalo (SUNY) *Lecture notes from Network Flows by Ahuja, Magnanti

### MATH 240 Fall, Chapter 1: Linear Equations and Matrices

MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS

### Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

### CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

### DATA ANALYSIS II. Matrix Algorithms

DATA ANALYSIS II Matrix Algorithms Similarity Matrix Given a dataset D = {x i }, i=1,..,n consisting of n points in R d, let A denote the n n symmetric similarity matrix between the points, given as where

### Math 4707: Introduction to Combinatorics and Graph Theory

Math 4707: Introduction to Combinatorics and Graph Theory Lecture Addendum, November 3rd and 8th, 200 Counting Closed Walks and Spanning Trees in Graphs via Linear Algebra and Matrices Adjacency Matrices

### 1 Polyhedra and Linear Programming

CS 598CSC: Combinatorial Optimization Lecture date: January 21, 2009 Instructor: Chandra Chekuri Scribe: Sungjin Im 1 Polyhedra and Linear Programming In this lecture, we will cover some basic material

### Minimize subject to. x S R

Chapter 12 Lagrangian Relaxation This chapter is mostly inspired by Chapter 16 of [1]. In the previous chapters, we have succeeded to find efficient algorithms to solve several important problems such

### Social Media Mining. Graph Essentials

Graph Essentials Graph Basics Measures Graph and Essentials Metrics 2 2 Nodes and Edges A network is a graph nodes, actors, or vertices (plural of vertex) Connections, edges or ties Edge Node Measures

### Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest.

2 Trees What is a tree? Forests and Trees: A forest is a graph with no cycles, a tree is a connected forest. Theorem 2.1 If G is a forest, then comp(g) = V (G) E(G). Proof: We proceed by induction on E(G).

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Lecture 9. 1 Introduction. 2 Random Walks in Graphs. 1.1 How To Explore a Graph? CS-621 Theory Gems October 17, 2012

CS-62 Theory Gems October 7, 202 Lecture 9 Lecturer: Aleksander Mądry Scribes: Dorina Thanou, Xiaowen Dong Introduction Over the next couple of lectures, our focus will be on graphs. Graphs are one of

### Why graph clustering is useful?

Graph Clustering Why graph clustering is useful? Distance matrices are graphs as useful as any other clustering Identification of communities in social networks Webpage clustering for better data management

### 6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

### Recall that two vectors in are perpendicular or orthogonal provided that their dot

Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

### Answers to some of the exercises.

Answers to some of the exercises. Chapter 2. Ex.2.1 (a) There are several ways to do this. Here is one possibility. The idea is to apply the k-center algorithm first to D and then for each center in D

### Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

### On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

### Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

### On Total Domination in Graphs

University of Houston - Downtown Senior Project - Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper

### Chapter 4: Trees. 2. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

9 Properties of Trees. Definitions: Chapter 4: Trees forest - a graph that contains no cycles tree - a connected forest. Theorem: Let T be a graph with n vertices. Then the following statements are equivalent:

### ECEN 5682 Theory and Practice of Error Control Codes

ECEN 5682 Theory and Practice of Error Control Codes Convolutional Codes University of Colorado Spring 2007 Linear (n, k) block codes take k data symbols at a time and encode them into n code symbols.

### Graph. Consider a graph, G in Fig Then the vertex V and edge E can be represented as:

Graph A graph G consist of 1. Set of vertices V (called nodes), (V = {v1, v2, v3, v4...}) and 2. Set of edges E (i.e., E {e1, e2, e3...cm} A graph can be represents as G = (V, E), where V is a finite and

### On Some Vertex Degree Based Graph Invariants

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (20) 723-730 ISSN 0340-6253 On Some Vertex Degree Based Graph Invariants Batmend Horoldagva a and Ivan

### Geometry of Linear Programming

Chapter 2 Geometry of Linear Programming The intent of this chapter is to provide a geometric interpretation of linear programming problems. To conceive fundamental concepts and validity of different algorithms

### MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3. University of Split, Croatia

GLASNIK MATEMATIČKI Vol. 38(58)(2003), 2 232 MIX-DECOMPOSITON OF THE COMPLETE GRAPH INTO DIRECTED FACTORS OF DIAMETER 2 AND UNDIRECTED FACTORS OF DIAMETER 3 Damir Vukičević University of Split, Croatia

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Steiner Tree and TSP Date: 01/29/15 Scribe: Katie Henry 2.1 Steiner Tree Definition 2.1.1 In the Steiner Tree problem the input

### Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques. My T. Thai

Approximation Algorithms: LP Relaxation, Rounding, and Randomized Rounding Techniques My T. Thai 1 Overview An overview of LP relaxation and rounding method is as follows: 1. Formulate an optimization

### Finding and counting given length cycles

Finding and counting given length cycles Noga Alon Raphael Yuster Uri Zwick Abstract We present an assortment of methods for finding and counting simple cycles of a given length in directed and undirected

### Graph Theory Notes. Vadim Lozin. Institute of Mathematics University of Warwick

Graph Theory Notes Vadim Lozin Institute of Mathematics University of Warwick 1 Introduction A graph G = (V, E) consists of two sets V and E. The elements of V are called the vertices and the elements

### Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

### Orthogonal Diagonalization of Symmetric Matrices

MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding

### Graph Theory. Introduction. Distance in Graphs. Trees. Isabela Drămnesc UVT. Computer Science Department, West University of Timişoara, Romania

Graph Theory Introduction. Distance in Graphs. Trees Isabela Drămnesc UVT Computer Science Department, West University of Timişoara, Romania November 2016 Isabela Drămnesc UVT Graph Theory and Combinatorics

### 15 Markov Chains: Limiting Probabilities

MARKOV CHAINS: LIMITING PROBABILITIES 67 Markov Chains: Limiting Probabilities Example Assume that the transition matrix is given by 7 2 P = 6 Recall that the n-step transition probabilities are given

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Notes on Determinant

ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without

### Kragujevac Trees and Their Energy

SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 6, 2 (2014), 71-79. Kragujevac Trees and Their Energy Ivan Gutman INVITED PAPER Abstract: The Kragujevac

### Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.2 Angles and the Dot Product Suppose x = (x 1, x 2 ) and y = (y 1, y 2 ) are two vectors in R 2, neither of which is the zero vector 0. Let α and

### Lecture 3: Linear Programming Relaxations and Rounding

Lecture 3: Linear Programming Relaxations and Rounding 1 Approximation Algorithms and Linear Relaxations For the time being, suppose we have a minimization problem. Many times, the problem at hand can

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### Why? A central concept in Computer Science. Algorithms are ubiquitous.

Analysis of Algorithms: A Brief Introduction Why? A central concept in Computer Science. Algorithms are ubiquitous. Using the Internet (sending email, transferring files, use of search engines, online

### Unicyclic Graphs with Given Number of Cut Vertices and the Maximal Merrifield - Simmons Index

Filomat 28:3 (2014), 451 461 DOI 10.2298/FIL1403451H Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Unicyclic Graphs with Given Number

### Inner Product Spaces and Orthogonality

Inner Product Spaces and Orthogonality week 3-4 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,

### Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition

Markov Chains, Stochastic Processes, and Advanced Matrix Decomposition Jack Gilbert Copyright (c) 2014 Jack Gilbert. Permission is granted to copy, distribute and/or modify this document under the terms

### Chapter 6. Orthogonality

6.3 Orthogonal Matrices 1 Chapter 6. Orthogonality 6.3 Orthogonal Matrices Definition 6.4. An n n matrix A is orthogonal if A T A = I. Note. We will see that the columns of an orthogonal matrix must be

### The Open University s repository of research publications and other research outputs

Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

### Linear Algebra and its Applications

Linear Algebra and its Applications 438 2013) 1393 1397 Contents lists available at SciVerse ScienceDirect Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa Note on the

### Degree Hypergroupoids Associated with Hypergraphs

Filomat 8:1 (014), 119 19 DOI 10.98/FIL1401119F Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Degree Hypergroupoids Associated

### Solution based on matrix technique Rewrite. ) = 8x 2 1 4x 1x 2 + 5x x1 2x 2 2x 1 + 5x 2

8.2 Quadratic Forms Example 1 Consider the function q(x 1, x 2 ) = 8x 2 1 4x 1x 2 + 5x 2 2 Determine whether q(0, 0) is the global minimum. Solution based on matrix technique Rewrite q( x1 x 2 = x1 ) =

### Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

### ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD. 1. Introduction

ON INDUCED SUBGRAPHS WITH ALL DEGREES ODD A.D. SCOTT Abstract. Gallai proved that the vertex set of any graph can be partitioned into two sets, each inducing a subgraph with all degrees even. We prove

### A graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices.

1 Trees A graph is called a tree, if it is connected and has no cycles. A star is a tree with one vertex adjacent to all other vertices. Theorem 1 For every simple graph G with n 1 vertices, the following

### DO NOT RE-DISTRIBUTE THIS SOLUTION FILE

Professor Kindred Math 04 Graph Theory Homework 7 Solutions April 3, 03 Introduction to Graph Theory, West Section 5. 0, variation of 5, 39 Section 5. 9 Section 5.3 3, 8, 3 Section 7. Problems you should

### Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

### Intersection Dimension and Maximum Degree

Intersection Dimension and Maximum Degree N.R. Aravind and C.R. Subramanian The Institute of Mathematical Sciences, Taramani, Chennai - 600 113, India. email: {nraravind,crs}@imsc.res.in Abstract We show

### Diameter and Treewidth in Minor-Closed Graph Families, Revisited

Algorithmica manuscript No. (will be inserted by the editor) Diameter and Treewidth in Minor-Closed Graph Families, Revisited Erik D. Demaine, MohammadTaghi Hajiaghayi MIT Computer Science and Artificial

### THREE DIMENSIONAL GEOMETRY

Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,

### 1 Plane and Planar Graphs. Definition 1 A graph G(V,E) is called plane if

Plane and Planar Graphs Definition A graph G(V,E) is called plane if V is a set of points in the plane; E is a set of curves in the plane such that. every curve contains at most two vertices and these

### Laplacian spectrum of weakly quasi-threshold graphs

Laplacian spectrum of weakly quasi-threshold graphs R. B. Bapat 1, A. K. Lal 2, Sukanta Pati 3 1 Stat-Math Unit, Indian Statistical Institute Delhi, 7-SJSS Marg, New Delhi - 110 016, India; e-mail: rbb@isid.ac.in

### Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency.

Mária Markošová Graph definition Degree, in, out degree, oriented graph. Complete, regular, bipartite graph. Graph representation, connectivity, adjacency. Isomorphism of graphs. Paths, cycles, trials.

### Section 6.1 - Inner Products and Norms

Section 6.1 - Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,