# A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS. Aysun Aytac 1, Betul Atay 2. Faculty of Science Ege University 35100, Bornova, Izmir, TURKEY

Save this PDF as:

Size: px
Start display at page:

Download "A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS. Aysun Aytac 1, Betul Atay 2. Faculty of Science Ege University 35100, Bornova, Izmir, TURKEY"

## Transcription

1 International Journal of Pure and Applied Mathematics Volume 03 No. 05, 6-70 ISSN: (printed version); ISSN: (on-line version) url: doi: PAijpam.eu A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS Aysun Aytac, Betul Atay, Department of Mathematics Faculty of Science Ege University 3500, Bornova, Izmir, TURKEY i j Abstract: Networks are known to be prone to node or link failures. A central issue in the analysis of complex networks is the assessment of their robustness and vulnerability. A network is usually represented by an undirected simple graph where vertices represent processors and edges represent links between processors. Different approaches to properly define a measure for graph vulnerability has been proposed so far. The distance d(i,j) between any two vertices i and j in a graph is the number of edges in a shortest path between i and j. If there is no path connecting i and j, then d(i,j) =. Latora and Marchiori introduced the measure of efficiency between vertices in a graph in 00. The unweighted efficiency between two vertices i and j is defined to be ε ij=/dij for all i j. The global efficiency of a graph E glob = n(n ) ε(v i,v j ) which is simply the average of the efficiencies over all pairs of distinct n vertices. In this paper, we study the global efficiency of special graphs. The behavior of this distance-related parameter on special graphs is taken into account by graphical analysis. AMS Subject Classification: 05C40, 68M0, 68R0 Key Words: graph vulnerability, network design and communication, efficiency, global efficiency Received: March 3, 05 c 05 Academic Publications, Ltd. url:

2 6 A. Aytac, B. Atay. Introduction Networks are used for modeling different systems such as chemical systems, neural networks, social systems or the Internet and the World Wide Web. The stability of a communication network, composed of processing nodes and communication links, is of prime importance to network designers. Vulnerability is a generalized conception. There are a lot of different definitions of vulnerability. Vulnerability is usually portrayed in negative terms as the susceptibility to be harmed. It is degree to which a systemic susceptible to and is unable to cope with adverse effects. In this paper, the vulnerability is defined as the decrease of system efficiency after attack. The vulnerability of a communication network measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. Communication networks must be constructed to be as stable as possible, not only with respect to the initial disruption, but also with respect to the possible reconstruction of the network. If we think of a graph as modeling a network, there have been several proposals for measures of the stability of a communication network including connectivity, toughness, scattering number, binding number and integrity (see []-[6]- [9]- [3]). In 00, Latora and Marchiori introduced the measure of efficiency between vertices in a graph (see [0]). The vulnerability of a network can be measured by the global efficiency of the graph describing the network. The global efficiency is a measure of the network performance and network efficiency based in the structure or topology of the network by the connections of the graph. The aim of global efficiency is simply to measure the average of the efficiencies over all pairs of the distinct vertices. Consider two different graph having the same graph vulnerability characteristics: their connectivity, toughness, integrity, binding number and scattering number is equal. In such a case, global efficiency recognizes the difference between these two graphs. In this paper, we consider simple finite undirected graphs without loops and multiple edges. Let G = (V,E) be a graph with a vertex set V=V(G) and an edge set E=E(G). The order of G is the number of vertices in G. The distance d(u,v) between two vertices u and v in G is the length of a shortest path between them. If u and v are not connected, then d(u,v) =, and for u=v, d(u,v)=0. The eccentricity of a vertex v in G is the distance from v to a vertex farthest away from v in G. The diameter of G, denoted by diam(g), is the largest distance between two vertices in V(G). The degree deg G (v) of a vertex v inv(g) is the number of edges incident to v. The maximum degree of G is (G) = {maxdeg G (v) v V(G)} (see [4]-[5]-[9]-[]-[3]).

3 A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS 63 The paper proceeds as follows. In Section, the definitions of efficiency and global efficiency and known results are given. Section 3 several results on global efficiency of some special graphs. Section 4 concludes the paper.. Efficiency and Global Efficiency In 00, Latora and Marchiori introduced the measure of efficiency between vertices in a graph ( see [0]). The global efficiency is a measure of the network performance and network efficiency based in the structure or topology of the network by the connections of the graph. The efficiency between two vertices i and j is defined to be ε(i,j) = d(i,j) for all i j, where d(i,j) is the length of a shortest path between two vertices i and j in G, so less distant connections are more valuable than more distant ones. The global efficiency of a graph E glob (G) = n(n ) ε(i,j) which is simply the average of the efficiencies i j V(G) over all pairs of the distinct vertices. Note that 0 E glob (G) is a normalized measure where its maximum value is reached for the complete graph K n on n nodes. In this sense, the global efficiency is expressed as a percentage of network performance, ε(i,j), i j V(G) of what ideally could be expected, n(n ), in a complete graph as ideal case. Since for a given number of nodes, the performance and therefore the global efficiency of the graph increases with the number of edges. Vulnerability measures the stability and robustness of the global performance of the graph under external perturbations (random or targeted failures or attacks). When vulnerability is defined as the relative drop in the global efficiency E glob (G), a negative value of vulnerability can appear (see[] -[3]-[7]- [8]-[]). As in examples, we consider the two graphs, both of which have same edges and vertices. For the connectivity and the diameter value of two graphs G and G the equalities are k(g )=k(g ) and diam(g )=diam(g ). According to these statements, we can not say that which graph is more reliable. Then, for measuring the reliability, we compute average global efficiency of these two graphs. If E glob (G ) < E glob (G ), we can say that graph G is more reliable than graph G. Theorem. [8] E glob (P n )= n n i n(n i. i= Theorem. [8] lim x E glob(p n ) = 0.

4 64 A. Aytac, B. Atay 3. Global Efficiency of Some Special Graphs In this section, we give some results on the global efficiency of some special graphs are calculated. These graphs are C n cycle graph, S,n star graph, W,n wheel graph, K n complete graph. Theorem 3. If C n is a cycle graph with n vertices, then the global efficiency of the cycle is E glob (C n ) = n/ n i + i= { (n ) n/ (n ) n/, if n is odd, if n is even Proof. There exist two cases for the shortest paths between the pairs of vertices according to diameter of the graph. Case If d(v i,v j ) < diam(c n ) for v i,v j V(C n ), then the length of shortest path in C n is k, where k =,diam(c n ). Hence, the value of efficiency of these vertices is ε(v i,v j ) = k, for all v i v j. Since this value of ε(i,j) holds for every other the pairs of vertices C n, combining the values for v i,v j V(C n ) yields diam(c n) k= ε(v i,v j ) = diam(c n) k= k. There are n different pairs of vertices in this case. Hence, we have diam(cn) k= diam(c n) k = n Case Ifd(v i,v j ) = diam(c n ) v i,v j V(C n ), thenwehavetwocasesdepending on the vertices of C n. Subcase Let n be odd. Then, we consider the pairs of vertices in such that d(v i,v i+diam(cn)) = diam(c n ), where i =,n, and the i+diam(c n ) is taken modula n. So, the value of efficiency for these vertices is ε(v i,v i+diam(cn)) = diam(c. It is clear that C n) n has diameter n. There are n different pairs of vertices in this case. Hence, we have, ε(vi,v i+ n ) = n n/. k= k.

5 A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS 65 By Case, Subcase of Case and the definition of global efficiency, the following result can be derived easily. ( ) where, n is odd. E glob (C n ) = n(n ) n n/ i= i +n n/ Subcase Let n be even. Then, we consider the pairs of vertices in such that d(v i,v i+ n/ ) = n/, where i =, n, and the i + n/ is taken modula n. Hence, the value of efficiency is ε(v i,v i+ n/ ) = n/. Since the value of ε(i,j) holds for every other n/ pairs of vertices C n, combining the efficiency values for v i,v j V(C n ) yields ε(vi,v i+ n ) = n n/ By Case, Subcase of Case and the definition of global efficiency, the following result can be derived easily. where, n is even. E glob (C n ) = n/ n(n ) (n By summing Case and, we obtain, i= i + n n/ ) The proof is completed. E glob (C n ) = n/ n i + i= { (n ) n/ (n ) n/, if n is odd, if n is even Theorem 4. If K n is a complete graph with n vertices, then the global efficiency of the complete graph is E glob (K n ) =. Proof. Since in a complete graph all vertices are mutually adjacent, distances between each pair of vertices are, that is, for v i,v j V(K n ), i j, i,j =,n, d(v i,v j ) =. By the definition of global efficiency, the following result can be derived easily. E glob (K n ) = n(n ) [ ( n ) ]

6 66 A. Aytac, B. Atay The proof is completed. = n(n ) = n(n ) Theorem 5. If W,n is a wheel graph with n+ vertices, then the global efficiency of the wheel graph is E glob (W,n ) = n+5 (n+). Proof. The vertices of W,n are of two kinds: n vertices which are of degree 3 will be referred to as minor vertices and the vertex of degree n will be referred to as central vertex. Label the minor vertices as v i (i =,n), the central vertex as c. There exist two cases for the shortest paths between the pairs of vertices. Case If the pair of vertices includes the central vertex and the minor vertices: There exists only one path between those vertices that has the length as d(c,v i ) =. Hence, the value of efficiency between these vertices is ε(c,v i ) =, i =,n. Since the value of ε(c,v i ) holds for every other n pairs of vertices W,n, combining the efficiency values for c,v i V(W,n ) yields ε(c,vi ) = n. Case If the pair of vertices includes any two different minor vertices v i and v j : We have three subcases for these minor vertices according to the length of the shortest path on the cycle between the vertices: Subcase If d(v i,v j ) =, the value of efficiency for these vertices is ε(v i,v j ) =. Since the value of ε(v i,v j ) holds for every other n pairs of vertices W,n, combining the efficiency values for v i,v j V(W,n ) yields ε(vi,v j ) = n. Subcase If d(v i,v j ) =, then the value of efficiency for these vertices is ε(v i,v j ) = /. Since the value of ε(v i,v j ) holds for every other n pairs of vertices W,n, combining the efficiency values for v i,v j V(W,n ) yields ε(vi,v j ) = n/.

7 A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS 67 Subcase 3 If d(v i,v j ) >, then there are only one path between the vertices v i and v j with length two including central vertex c. Hence, the value of efficiency for these vertices is ε(v i,v j ) = /. Since the value of ε(v i,v j ) holds for every other n(n 5)/ pairs of vertices W,n, combining the efficiency values for v i,v j V(W,n ) yields ε(vi,v j ) = n(n 5)/4. Consequently, by summing up the cases and, the value of the global efficiency of V(W,n ) is The proof is completed. E glob(w,n ) = = ( n+n+ n n(n+) + n(n 5) ) 4 n+5 (n+). Theorem 6. If S,n is a star graph with n + vertices, then the global efficiency of the star graph is E glob (S,n ) = n+3 (n+). Proof. The vertices of S,n are of two kinds: one vertex of degree n and n vertices of degree one. The vertices of degree one will be referred to as minor vertices as v i (i =,n) and vertex of degree n to as center vertex as c. There exist two cases for the shortest paths between the pairs of vertices. Case If d(v i,v j ) =, then this case exists for only the pairs of vertices including the central vertex and the minor vertices. Hence, the value of efficiency for these vertices is ε(c,v i ) = i =,n.. Since the value of ε(c,v i ) holds for every other n pairs of vertices S,n, combining the efficiency values for c,v i V(S,n ) yields ε(c,vi ) = n. Case If d(v i,v j ) =, then this case exists for only the pairs of vertices including the different two minor vertices. Hence, the value of efficiency for these vertices is ε(v i,v j ) = /, i,j =,n,i j. Since the value of ε(v i,v j ) holds for every other ( n ) = n(n )/ pairs of vertices S,n, combining the efficiency values for c,v i V(S,n ) yields ε(c,vi ) = n(n )/4.

8 68 A. Aytac, B. Atay Consequently, by summing up the cases and, the value of the global efficiency of S,n is E glob (S,n ) = = n(n ) (n+ ) n(n+) 4 n+3 (n+) The proof is completed. 4. Conclusion The global efficiencies of graphs discussed in the paper are given in Table. Table : The global efficiency values of some graphs Graph Vertex Number Edge Number Global Efficiency, if n is odd (n ) n/ C n cycle graph n n, if n is even (n ) n/ ( K n complete graph n n ) = n(n ) n+5 W,n wheel graph n+ n (n+) S,n star graph n+ n n+3 (n+) The line diagram for n =,5 is gained above Figure by using the values obtained in Table. Accordingly, it is possible to make a comparison between some known special graph types discussed in the paper. A comparison is made in order to decide which type is more stable with respect to the line diagram obtained in Figure. According to the data obtained, it can be observed that the order for the graphs from more stable to less stable is complete graph, wheel, star, cycle and path.

9 A MEASURE OF GLOBAL EFFICIENCY IN NETWORKS 69 Figure : The comparison about the global efficiency of some special graphs References [] C.A.Barefoot, R.Entringer, and H.Swart, Vulnerability in Graphs-A Comparative Survey. J.Comb.Math.Comb.Comput., (987), 3-. [] Boccaletti S., Buldu J.,Criado R.,Flores J.,Latora V.,Pello J. and Romance M., The Structure and Dynamics of Complex Networks, Chaos 7, 0430 (007). [3] Boccaletti S., Criado R.,Pello J., Romance M. and Vela-Perez M., Vulnerability and fall of efficiency in complex networks: A new approach with computational advantages, Int. J. Bifurcation Chaos 9,77-735(009). [4] Bondy J.A. and Murty U.S.R., Graph theory with applications, American Elsevier Publishing Co., Inc., New York(976). [5] Chartrand G., Leisnak L., Graphs and Digraphs, Second Edition, Wadsworth.Monterey (986) [6] V.Chvatal, Tough Graphs and Hamiltonian Circuits., Discrete Math., 5, (973), 5-8. [7] Crucitti P., Latora V., Marchiori M. and Rapisarda A., Error and attacktolerance of complex networks, Physica A 340, (004).

10 70 A. Aytac, B. Atay [8] Ek B.,Caitlin V. and Narayan D.A., Efficiency of star-like graphs and the Atlanta subway network,phisica A 39, (03). [9] Jung H. A., On a class of posets and the corresponding comparability graphs, Journal of Combinatorial Theory, Series B 4() (978) [0] Latora V. and Marchiori M., Efficient behavior of small-world networks, Phys.Rev. Lett. 87,9870,00. [] Latora V. and Marchiori M., A measure of centrality based on network efficiency, New Journal of Phys. 9,88(007). [] West D.B., Introduction to Graph Theory, Prentice Hall, NJ(00). [3] Woodall D.R., The binding number of a graph and its Anderson number, J. Combin. Theory Ser. B 5 (973) 5-55.

### BOUNDARY EDGE DOMINATION IN GRAPHS

BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 0-4874, ISSN (o) 0-4955 www.imvibl.org /JOURNALS / BULLETIN Vol. 5(015), 197-04 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA

### CSE 20: Discrete Mathematics for Computer Science. Prof. Miles Jones. Today s Topics: Graphs. The Internet graph

Today s Topics: CSE 0: Discrete Mathematics for Computer Science Prof. Miles Jones. Graphs. Some theorems on graphs. Eulerian graphs Graphs! Model relations between pairs of objects The Internet graph!

### Tenacity and rupture degree of permutation graphs of complete bipartite graphs

Tenacity and rupture degree of permutation graphs of complete bipartite graphs Fengwei Li, Qingfang Ye and Xueliang Li Department of mathematics, Shaoxing University, Shaoxing Zhejiang 312000, P.R. China

### A Study of Sufficient Conditions for Hamiltonian Cycles

DeLeon 1 A Study of Sufficient Conditions for Hamiltonian Cycles Melissa DeLeon Department of Mathematics and Computer Science Seton Hall University South Orange, New Jersey 07079, U.S.A. ABSTRACT A graph

### 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

### 1 Digraphs. Definition 1

1 Digraphs Definition 1 Adigraphordirected graphgisatriplecomprisedofavertex set V(G), edge set E(G), and a function assigning each edge an ordered pair of vertices (tail, head); these vertices together

### 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

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis. Contents. Introduction. Maarten van Steen. Version: April 28, 2014

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R.0, steen@cs.vu.nl Chapter 0: Version: April 8, 0 / Contents Chapter Description 0: Introduction

### Network (Tree) Topology Inference Based on Prüfer Sequence

Network (Tree) Topology Inference Based on Prüfer Sequence C. Vanniarajan and Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology Madras Chennai 600036 vanniarajanc@hcl.in,

### Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

### Graph Theory and Complex Networks: An Introduction. Chapter 06: Network analysis

Graph Theory and Complex Networks: An Introduction Maarten van Steen VU Amsterdam, Dept. Computer Science Room R4.0, steen@cs.vu.nl Chapter 06: Network analysis Version: April 8, 04 / 3 Contents Chapter

### 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.

### 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

### Special Classes of Divisor Cordial Graphs

International Mathematical Forum, Vol. 7,, no. 35, 737-749 Special Classes of Divisor Cordial Graphs R. Varatharajan Department of Mathematics, Sri S.R.N.M. College Sattur - 66 3, Tamil Nadu, India varatharajansrnm@gmail.com

### Topological Analysis of the Subway Network of Madrid

Topological Analysis of the Subway Network of Madrid Mary Luz Mouronte Departamento de Ingeniería y Arquitecturas Telemáticas. Escuela Técnica Superior de Ingeniería y Sistemas detelecomunicación Universidad

### [f(u)] + [f(v)] {0, 1, 2, 3,...,2p } such that when each edge uv is assigned the label f(uv) =

ISSN: -6 ISO :8 Certified Volume, Issue, May A Study on Root Mean Square Labelings in Graphs *P. Shalini & **D. Paul Dhayabaran *Asst Professor, Cauvery College for Women,**Principal, Bishop Heber College

### 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

### The Metric Dimension of Graph with Pendant Edges

The Metric Dimension of Graph with Pendant Edges H. Iswadi 1,2, E.T. Baskoro 2, R. Simanjuntak 2, A.N.M. Salman 2 2 Combinatorial Mathematics Research Group Faculty of Mathematics and Natural Science,

### About the Tutorial. Audience. Prerequisites. Disclaimer & Copyright

About the Tutorial This tutorial offers a brief introduction to the fundamentals of graph theory. Written in a reader-friendly style, it covers the types of graphs, their properties, trees, graph traversability,

### 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

### 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.

### Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

### P. Jeyanthi and N. Angel Benseera

Opuscula Math. 34, no. 1 (014), 115 1 http://dx.doi.org/10.7494/opmath.014.34.1.115 Opuscula Mathematica A TOTALLY MAGIC CORDIAL LABELING OF ONE-POINT UNION OF n COPIES OF A GRAPH P. Jeyanthi and N. Angel

### 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.

### A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE. 1. Introduction and Preliminaries

Acta Math. Univ. Comenianae Vol. LXVI, 2(1997), pp. 285 291 285 A REMARK ON ALMOST MOORE DIGRAPHS OF DEGREE THREE E. T. BASKORO, M. MILLER and J. ŠIRÁŇ Abstract. It is well known that Moore digraphs do

### 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

### Divisor graphs have arbitrary order and size

Divisor graphs have arbitrary order and size arxiv:math/0606483v1 [math.co] 20 Jun 2006 Le Anh Vinh School of Mathematics University of New South Wales Sydney 2052 Australia Abstract A divisor graph G

### SCORE SETS IN ORIENTED GRAPHS

Applicable Analysis and Discrete Mathematics, 2 (2008), 107 113. Available electronically at http://pefmath.etf.bg.ac.yu SCORE SETS IN ORIENTED GRAPHS S. Pirzada, T. A. Naikoo The score of a vertex v in

### Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages

Frans J.C.T. de Ruiter, Norman L. Biggs Applications of integer programming methods to cages Article (Published version) (Refereed) Original citation: de Ruiter, Frans and Biggs, Norman (2015) Applications

### SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH

CHAPTER 3 SEMITOTAL AND TOTAL BLOCK-CUTVERTEX GRAPH ABSTRACT This chapter begins with the notion of block distances in graphs. Using block distance we defined the central tendencies of a block, like B-radius

### Extremal Wiener Index of Trees with All Degrees Odd

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 70 (2013) 287-292 ISSN 0340-6253 Extremal Wiener Index of Trees with All Degrees Odd Hong Lin School of

### 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

### Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

### 3. Eulerian and Hamiltonian Graphs

3. Eulerian and Hamiltonian Graphs There are many games and puzzles which can be analysed by graph theoretic concepts. In fact, the two early discoveries which led to the existence of graphs arose from

### Some Polygonal Sum Labeling of Paths

Volume 6 No.5, January 03 Some Polygonal Sum Labeling of Paths S.Murugesan Associate Professor CBM college Coimbatore-60 D.Jayaraman Associate Professor CBM College Coimbatore- 60 J.Shiama Assistant Professor

### Product irregularity strength of certain graphs

Also available at http://amc.imfm.si ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 7 (014) 3 9 Product irregularity strength of certain graphs Marcin Anholcer

### 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.

### Available online at ScienceDirect. Procedia Computer Science 74 (2015 ) 47 52

Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 74 (05 ) 47 5 International Conference on Graph Theory and Information Security Fractional Metric Dimension of Tree and

### Degree-associated reconstruction parameters of complete multipartite graphs and their complements

Degree-associated reconstruction parameters of complete multipartite graphs and their complements Meijie Ma, Huangping Shi, Hannah Spinoza, Douglas B. West January 23, 2014 Abstract Avertex-deleted subgraphofagraphgisacard.

### Social Media Mining. Network Measures

Klout Measures and Metrics 22 Why Do We Need Measures? Who are the central figures (influential individuals) in the network? What interaction patterns are common in friends? Who are the like-minded users

### Distance Two Vertex-Magic Graphs

Distance Two Verte-Magic Graphs Ebrahim Salehi Department of Mathematical Sciences Universit of Nevada Las Vegas Las Vegas, NV 8954-4020 ebrahim.salehi@unlv.edu Abstract Given an abelian group A, a graph

### Edge looseness of plane graphs

Also available at http://amc-journal.eu ISSN 1855-966 (printed edn.), ISSN 1855-974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (015) 89 96 Edge looseness of plane graphs Július Czap Department of

### . 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

### 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

### Total colorings of planar graphs with small maximum degree

Total colorings of planar graphs with small maximum degree Bing Wang 1,, Jian-Liang Wu, Si-Feng Tian 1 Department of Mathematics, Zaozhuang University, Shandong, 77160, China School of Mathematics, Shandong

### Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

### A note on properties for a complementary graph and its tree graph

A note on properties for a complementary graph and its tree graph Abulimiti Yiming Department of Mathematics Institute of Mathematics & Informations Xinjiang Normal University China Masami Yasuda Department

### The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson)

Chapter 11 Graph Theory The origins of graph theory are humble, even frivolous. Biggs, E. K. Lloyd, and R. J. Wilson) (N. Let us start with a formal definition of what is a graph. Definition 72. A graph

### Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010. Chapter 7: Digraphs

MCS-236: Graph Theory Handout #Ch7 San Skulrattanakulchai Gustavus Adolphus College Dec 6, 2010 Chapter 7: Digraphs Strong Digraphs Definitions. A digraph is an ordered pair (V, E), where V is the set

### Generalizing the Ramsey Problem through Diameter

Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted: January 8, 001; Accepted: November 13, 001. MR Subject Classifications: 05C1, 05C15, 05C35, 05C55 Abstract Given a graph G and positive

### 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

### International Electronic Journal of Pure and Applied Mathematics IEJPAM, Volume 9, No. 1 (2015)

International Electronic Journal of Pure and Applied Mathematics Volume 9 No. 1 2015, 21-27 ISSN: 1314-0744 url: http://www.e.ijpam.eu doi: http://dx.doi.org/10.12732/iejpam.v9i1.3 THE ISOTROPY GRAPH OF

### 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.

### Tools for parsimonious edge-colouring of graphs with maximum degree three. J.L. Fouquet and J.M. Vanherpe. Rapport n o RR-2010-10

Tools for parsimonious edge-colouring of graphs with maximum degree three J.L. Fouquet and J.M. Vanherpe LIFO, Université d Orléans Rapport n o RR-2010-10 Tools for parsimonious edge-colouring of graphs

### 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

### V. Adamchik 1. Graph Theory. Victor Adamchik. Fall of 2005

V. Adamchik 1 Graph Theory Victor Adamchik Fall of 2005 Plan 1. Basic Vocabulary 2. Regular graph 3. Connectivity 4. Representing Graphs Introduction A.Aho and J.Ulman acknowledge that Fundamentally, computer

### Triangle deletion. Ernie Croot. February 3, 2010

Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,

### Graphs and Network Flows IE411 Lecture 1

Graphs and Network Flows IE411 Lecture 1 Dr. Ted Ralphs IE411 Lecture 1 1 References for Today s Lecture Required reading Sections 17.1, 19.1 References AMO Chapter 1 and Section 2.1 and 2.2 IE411 Lecture

### 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

### Discrete Applied Mathematics. The firefighter problem with more than one firefighter on trees

Discrete Applied Mathematics 161 (2013) 899 908 Contents lists available at SciVerse ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam The firefighter problem with

### 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

### Graph theoretic approach to analyze amino acid network

Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 31-37 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Graph theoretic approach to

### 8. Matchings and Factors

8. Matchings and Factors Consider the formation of an executive council by the parliament committee. Each committee needs to designate one of its members as an official representative to sit on the council,

### 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

### Hosoya Polynomials of Pentachains

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun Math Comput Chem 65 (0) 807-89 ISSN 0340-653 Hosoya Polynomials of Pentachains Ali A Ali Academy- University of Mosul Ahmed M

### Impact of Remote Control Failure on Power System Restoration Time

Impact of Remote Control Failure on Power System Restoration Time Fredrik Edström School of Electrical Engineering Royal Institute of Technology Stockholm, Sweden Email: fredrik.edstrom@ee.kth.se Lennart

### GRAPHS AND ZERO-DIVISORS. In an algebra class, one uses the zero-factor property to solve polynomial equations.

GRAPHS AND ZERO-DIVISORS M. AXTELL AND J. STICKLES In an algebra class, one uses the zero-factor property to solve polynomial equations. For example, consider the equation x 2 = x. Rewriting it as x (x

### Tree Cover of Graphs

Applied Mathematical Sciences, Vol. 8, 2014, no. 150, 7469-7473 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.49728 Tree Cover of Graphs Rosalio G. Artes, Jr. and Rene D. Dignos Department

### Simple Graphs Degrees, Isomorphism, Paths

Mathematics for Computer Science MIT 6.042J/18.062J Simple Graphs Degrees, Isomorphism, Types of Graphs Simple Graph this week Multi-Graph Directed Graph next week Albert R Meyer, March 10, 2010 lec 6W.1

### Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices

MATHEMATICAL COMMUNICATIONS 47 Math. Commun., Vol. 15, No. 2, pp. 47-58 (2010) Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices Hongzhuan Wang 1, Hongbo Hua 1, and Dongdong Wang

### Degree diameter problem on triangular networks

AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 63(3) (2015), Pages 333 345 Degree diameter problem on triangular networks Přemysl Holub Department of Mathematics University of West Bohemia P.O. Box 314,

### 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:

### CMSC 451: Graph Properties, DFS, BFS, etc.

CMSC 451: Graph Properties, DFS, BFS, etc. Slides By: Carl Kingsford Department of Computer Science University of Maryland, College Park Based on Chapter 3 of Algorithm Design by Kleinberg & Tardos. Graphs

### 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

### 8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

### 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

### Outline. NP-completeness. When is a problem easy? When is a problem hard? Today. Euler Circuits

Outline NP-completeness Examples of Easy vs. Hard problems Euler circuit vs. Hamiltonian circuit Shortest Path vs. Longest Path 2-pairs sum vs. general Subset Sum Reducing one problem to another Clique

### ON SUPER (a, d)-edge ANTIMAGIC TOTAL LABELING OF CERTAIN FAMILIES OF GRAPHS

Discussiones Mathematicae Graph Theory 3 (01) 535 543 doi:10.7151/dmgt.163 ON SUPER (a, d)-edge ANTIMAGIC TOTAL LABELING OF CERTAIN FAMILIES OF GRAPHS P. Roushini Leely Pushpam Department of Mathematics

### 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

### Basic Notions on Graphs. Planar Graphs and Vertex Colourings. Joe Ryan. Presented by

Basic Notions on Graphs Planar Graphs and Vertex Colourings Presented by Joe Ryan School of Electrical Engineering and Computer Science University of Newcastle, Australia Planar graphs Graphs may be drawn

### COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

### 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

### Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

### On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

### YILUN SHANG. e λi. i=1

LOWER BOUNDS FOR THE ESTRADA INDEX OF GRAPHS YILUN SHANG Abstract. Let G be a graph with n vertices and λ 1,λ,...,λ n be its eigenvalues. The Estrada index of G is defined as EE(G = n eλ i. In this paper,

### 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.,

### Mean Ramsey-Turán numbers

Mean Ramsey-Turán numbers Raphael Yuster Department of Mathematics University of Haifa at Oranim Tivon 36006, Israel Abstract A ρ-mean coloring of a graph is a coloring of the edges such that the average

### FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS. Mateusz Nikodem

Opuscula Mathematica Vol. 32 No. 4 2012 http://dx.doi.org/10.7494/opmath.2012.32.4.751 FALSE ALARMS IN FAULT-TOLERANT DOMINATING SETS IN GRAPHS Mateusz Nikodem Abstract. We develop the problem of fault-tolerant

### Communicated by Bijan Taeri. 1. Introduction

Transactions on Combinatorics ISSN (print: 51-8657, ISSN (on-line: 51-8665 Vol. 3 No. 3 (01, pp. 3-9. c 01 University of Isfahan www.combinatorics.ir www.ui.ac.ir STUDYING THE CORONA PRODUCT OF GRAPHS

### Types of Degrees in Bipolar Fuzzy Graphs

pplied Mathematical Sciences, Vol. 7, 2013, no. 98, 4857-4866 HIKRI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.37389 Types of Degrees in Bipolar Fuzzy Graphs Basheer hamed Mohideen Department

### Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

### SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM

SPERNER S LEMMA AND BROUWER S FIXED POINT THEOREM ALEX WRIGHT 1. Intoduction A fixed point of a function f from a set X into itself is a point x 0 satisfying f(x 0 ) = x 0. Theorems which establish the

### 14. Trees and Their Properties

14. Trees and Their Properties This is the first of a few articles in which we shall study a special and important family of graphs, called trees, discuss their properties and introduce some of their applications.

### 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

### Midterm Practice Problems

6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

### Network/Graph Theory. What is a Network? What is network theory? Graph-based representations. Friendship Network. What makes a problem graph-like?

What is a Network? Network/Graph Theory Network = graph Informally a graph is a set of nodes joined by a set of lines or arrows. 1 1 2 3 2 3 4 5 6 4 5 6 Graph-based representations Representing a problem

### A simple criterion on degree sequences of graphs

Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree

### Section Summary. Introduction to Graphs Graph Taxonomy Graph Models

Chapter 10 Chapter Summary Graphs and Graph Models Graph Terminology and Special Types of Graphs Representing Graphs and Graph Isomorphism Connectivity Euler and Hamiltonian Graphs Shortest-Path Problems