# On Maximal Energy of Trees with Fixed Weight Sequence

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 75 (2016) ISSN On Maximal Energy of Trees with Fixed Weight Sequence Shicai Gong 1,,, Xueliang Li 2,, Ivan Gutman 3,4, Guanghui Xu 1,, Yuxiang Tang 1, Zhongmei Qin 2, Kang Yang 2 1 School of Science, Zhejiang University of Science and Technology, Hangzhou, , P. R. China 2 Center for Combinatorics and LPMC TJKLC Nankai University, Tianjin , P. R. China 3 Faculty of Science, University of Kragujevac, P. O. Box 60, Kragujevac, Serbia 4 State University of Novi Pazar, Novi Pazar, Serbia (Received May 12, 2015) Abstract Let W = {w 1, w 2,..., w n 1 } be a sequence of positive numbers. Denote by T (n, W ) the set of all weighted trees of order n with weight sequence W. We show that a weighted tree that has maximal energy in T (n, W ), is a weighted path. For n 6, we determine the unique path having maximal energy in T (n, W ). For n 7, we give a conjecture on the structure and distribution of weights of the unique maximum energy tree in T (n, W ). Some results supporting this conjecture are obtained. Corresponding author Supported by the Zhejiang Provincial Natural Science Foundation of China, (No. LY12A01016) Supported by the National Natural Science Foundation of China, (No ) Supported by the National Natural Science Foundation of China, (No )

2 Introduction In this paper we consider trees on n vertices, to each edge of which a positive weight is assigned. The sequence of weights of the edges in a weighted tree is referred to as its weight sequence. Let W = {w 1, w 2,..., w n 1 } be a sequence of positive numbers. Denote by T (n, W ) the set of all weighted trees of order n with weight sequence W. In general, the energy of a weighted graph G of order n is defined as n E(G) = λ i where λ 1, λ 2,..., λ n are the (real) eigenvalues of the (nonnegative, symmetric) adjacency matrix A of G. More information on (weighted) graph energy can be found in [2 6, 10, 12, 15, 17, 21]. In [3], Brualdi et al. investigated the extremal energy of a class of integral weighted graphs. Let T (n, m) be the set of all weighted trees of order n with the fixed total weight sum m. They stated the following conjecture pertaining to the maximal energy of all integral weighted trees with fixed weight: i=1 Conjecture 1. [3, Conjecture 10] Let n 5 and m n. The path with weight sequence {m n + 2, 1,..., 1}, where the weight of one of the pendent edges equals m n + 2, is the unique integral weighted tree in T (n, m) with maximal energy. Let Ê(n, W ) = max{e(t ) : T T (n, W )} be the maximal energy of a tree in T (n, W ). Motivated by the above conjecture, in this paper we consider the following problem: Problem 2. For a given weight sequence W, determine the tree(s) in T (n, W ) whose energy achieves Ê(n, W ). We show that in T (n, W ), a weighted tree with maximal energy is a weighted path. Further, for small order n ( 6), we determine the unique path having maximal energy in T (n, W ). For larger order n, we give a conjecture on the structure and the distribution of weights of the unique tree in T (n, W ). Finally, some results supporting this conjecture are obtained.

3 Preliminary results We first introduce some terminology and notation. Let G = (V (G), E(G)) be a graph. For V 1 = {v 1, v 2,..., v s } V (G) and E 1 = {e 1, e 2,..., e k } E(G), denote by G\E 1 the graph obtained from G by deleting all edges of E 1 and by G\V 1 the graph obtained from G by removing all vertices of V 1 together with all incident edges. For convenience, we sometimes write G\e 1 e 2... e k and G\v 1 v 2... v s instead of G\E 1 and G\V 1, respectively. Denote by P n = u 1 e 1 u 2... u n 1 e n 1 u n the path on n vertices, where u i and u i+1 are the two endvertices of the edge e i. For convenience, we sometimes denote by P = u 1 w 1 u 2 w 2 u 3... u n 1 w n 1 u n or P = w 1 w 2... w n 1 a weighted path on n vertices, where w i denotes the weight of the edge e i for i = 1, 2,..., n 1. We refer to Cvetković et al. [5] for terminology and notation not defined here. A graph is said to be elementary if it is isomorphic either to P 2 or to a cycle. The weight of P 2 is defined as the square of the weight of its unique edge. The weight of a cycle is the product of the weights of all its edges. A graph H is called a Sachs graph if each component of H is an elementary graph [8, 9, 14]. The weight of a Sachs graph H, denoted by W (H ), is the product of the weights of all elementary subgraphs contained in H. Denote by φ(g, λ) the characteristic polynomial of a graph G, defined as φ(g, λ) = det [ λ I n A(G) ] n = a k (G) λ n k (1) where A(G) is the adjacency matrix of G and I n the identity matrix of order n. The following well known result determines all coefficients of the characteristic polynomial of a weighted graph in terms of its Sachs subgraphs [1, 5, 7, 19, 20]. Theorem 3. Let G be a weighted graph on n vertices with adjacency matrix A(G) and characteristic polynomial φ(g, λ) = n a k (G) λ n k. Then k=0 k=0 a i (G) = H ( 1) p(h ) 2 c(h ) W (H ) where the summation is over all Sachs subgraphs H of G having i vertices, and where p(h ) and c(h ) are, respectively, the number of components and the number of cycles contained in H.

4 -270- Similarly, we have the following recursions for the characteristic polynomial of a weighted graph [11]. Theorem 4. Let G be a weighted graph and e = uv be a cut edge of G. Denote by w(e) the weight of the edge e. Then φ(g, λ) = φ(g\e, λ) w(e) 2 φ(g\uv, λ). From the Coulson integral formula for the energy (see [4,15 17] and the references cited therein), it can be shown [10] that if G is a weighted bipartite graph with characteristic polynomial as in Eq. (1), then E(G) = 1 + n/2 λ 2 ln b 2k λ 2k dλ π where b k = a k. It follows that in the case of weighted trees, E(T ) is a strict monotonically increasing function of the numbers b 2i, i = 1, 2,..., n/2. Thus, in analogy to comparing the energies of two non-weighted trees [10, 22, 23], we introduce a quasi-ordering relation for weighted trees (see also [13, 18]): Definition 5. Let T 1 and T 2 be two weighted trees of order n. If b 2k (T 1 ) b 2k (T 2 ) for all k with 0 k n/2, then we write T 1 T 2. Furthermore, if T 1 T 2 and there exists at least one index k such that b 2k (T 1 ) < b 2k (T 2 ), then we write T 1 T 2. If b 2k (T 1 ) = b 2k (T 2 ) for all k, then we write T 1 T 2. Note that there are non-isomorphic weighted graphs T 1 and T 2 with T 1 T 2, which implies that in the general case is a quasi-ordering, but not a partial ordering. According to the integral formula above, we have for two weighted trees T 1 and T 2 of order n that T 1 T 2 = E(T 1 ) E(T 2 ) and T 1 T 2 = E(T 1 ) < E(T 2 ). (2) k=0

5 Main results An immediate consequence of Theorem 4 is: Corollary 6. Let G be a weighted bipartite graph with a cut edge e = uv. Suppose that the weight of the edge e is w e. Then b k (G) = b k (G\e) + we 2 b k 2 (G\uv) where b k = a k, as defined as above. Applying Corollary 6, we can deduce that a weighted tree with maximal energy in T (n, W ) is a weighted path. Theorem 7. For n 2, let W = {w 1, w 2,..., w n 1 } be a sequence of positive numbers. Let T be a weighted tree with maximal energy in T (n, W ). Then, T is a weighted path. Proof. We prove the theorem by induction on n. Obviously, the result holds for n = 2, 3. Let n 4 and suppose that the result is true for trees of orders less than n. For any n 4, suppose that u 1 is a pendent vertex of T and u 1 u 2 = e 1 has weight w e. Then using Corollary 6, we have b k (T ) = b k (T \u 1 ) + w 2 e b k 2 (T \u 1 u 2 ). Let W 1 be the weight sequence of W \w e and W 2 the weight sequence obtained from W by removal of the weights of all edges incident with u 2. Then by the induction assumption, the trees achieving Ê(n 1, W 1) and Ê(n 2, W 2) are paths or union of paths. Moreover, in order to maximize Ê(n 2, W 2), T \u 1 u 2 has to have n 3 edges. Consequently, both of T \u 1 and T \u 1 u 2 are paths and hence the result follows. In view of Theorem 7, in order to determine the tree(s) having energy Ê(n, W ), we focus on discussing the weight distribution of the path P n. For convenience, denote by P(n, W ) the set of all weighted paths of order n with weight sequence W. Lemma 8. For n 4, let P = u 1 e 1 u 2 e 2... u n 1 e n 1 u n P(n, W ). Let the weight of the edge e i be denoted by w i for i = 1, 2,..., n 1. If E(P ) = Ê(n, W ), then (a) w 1 w 2, and (b) w n 1 w n 2.

6 E(P ) = Ê(n, W ), we have E(P ) E(P ). (3) Proof. (a) Let P be the weighted path obtained from P by exchanging the edges e 1 and e 2, that is, P = u 1 e 2 u 2 e 1 u 3... u n 1 e n 1 u n. Then by the hypothesis that From Corollary 6, b k (P ) = b k (P \e 1 ) + w 2 1 b k 2 (P \u 1 u 2 ) and = b k (P \e 1 e 2 ) + w 2 2 b k 2 (P \u 1 u 2 u 3 ) + w 2 1 b k 2 (P \e 1 e 2 ) b k (P ) = b k (P \e 2 ) + w2 2 b k 2 (P \u 1 u 2 ) = b k (P \e 1 e 2 ) + w1 2 b k 2 (P \u 1 u 2 u 3 ) + w2 2 b k 2 (P \e 1 e 2 ). Assume to the contrary that w 1 < w 2. Note that P \u 1 u 2 u 3 = P \u 1 u 2 u 3, P \e 1 e 2 = P \e 1 e 2, and that P \u 1 u 2 u 3 is a proper subgraph of P \e 1 e 2. Then b k (P ) b k (P ) = (w1 2 w2) [ 2 b k 2 (P \e 1 e 2 ) b k 2 (P \u 1 u 2 u 3 ) ] 0 and there exists at least one index k such that b k (P ) b k (P ) < 0 as P \e 1 e 2 contains at least one edge. Thus P P, a contradiction to conditions (2) and (3). Consequently, the result (a) follows. The proof of (b) is analogous. Lemma 9. For n 5, let P = u 1 e 1 u 2 e 2 u 3 e 3 u 4... u n 1 e n 1 u n P(n, W ). Let the weight of the edge e i be denoted by w i for i = 1, 2,..., n 1. If E(P ) = Ê(n, W ), then (a) w 1 w 3, and (b) w n 1 w n 3. Proof. (a) Let P be the weighted path obtained from P by exchanging the edges e 1 and e 3, that is, P = u 1 e 3 u 2 e 2 u 3 e 1 u 4... u n 1 e n 1 u n. Then E(P ) E(P ). (4) By Corollary 6, b k (P ) = b k (P \e 1 ) + w1 2 b k 2 (P \u 1 u 2 ) = b k (P \e 1 e 3 ) + w3 2 b k 2 (P \u 2 u 3 u 4 ) + w1 2 b k 2 (P \u 1 u 2 u 3 ) + w1 2 w3 2 b k 4 (P \u 1 u 2 u 3 u 4 )

7 -273- and b k (P ) = b k (P \e 3 ) + w3 2 b k 2 (P \u 1 u 2 ) = b k (P \e 3 e 1 ) + w1 2 b k 2 (P \u 2 u 3 u 4 ) + w3 2 b k 2 (P \u 1 u 2 u 3 ) + w1 2 w3 2 b k 4 (P \u 1 u 2 u 3 u 4 ). Assume to the contrary that w 1 < w 3. Note that P \u 1 u 2 u 3 = P \u 1 u 2 u 3 and P \u 2 u 3 u 4 = P \u 2 u 3 u 4, and that P \u 2 u 3 u 4 is the union of the isolated vertex u 1 and a proper subgraph of P \u 1 u 2 u 3. Then b k (P ) b k (P ) = (w3 2 w1) [ 2 b k 2 (P \u 2 u 3 u 4 ) b k 2 (P \u 1 u 2 u 3 ) ] 0 and there exists at least one integer k such that b k (P ) b k (P ) < 0. Thus P P, a contradiction to condition (4). Consequently, the result (a) follows. The proof of (b) is analogous. For small order n, applying Lemmas 8 and 9, we can determine the paths having maximal energy among the weighted paths in P(n, W ). Theorem 10. For n 6, let W = {w 1, w 2,..., w n 1 } be a sequence of positive numbers such that w 1 w 2 w n 1, and let P = u 1 e 1 u 2 e 2... u n 1 e n 1 u n P(n, W ) be the path having energy Ê(n, W ). Suppose w(e 1) w(e n 1 ). Then for i = 1, 2,..., n 1, w i w(e i ) = otherwise. w n+1 i if either i n 1 2 n 1 and i is odd, or i > and n i is even, 2 Proof. Bearing in mind Lemma 8 and the hypothesis that w(e 1 ) w(e n 1 ), the result follows for n 4. For n = 5, from Lemmas 8 and 9, we have that the desired path is either P 1 = w 1 w 3 w 4 w 2 or P 2 = w 1 w 4 w 3 w 2. By a direct calculation, we have b 2 (P 1 ) = b 2 (P 2 ) b 4 (P 1 ) = w1 2 w4 2 + w1 2 w2 2 + w3 2 w2 2 b 4 (P 2 ) = w1 2 w3 2 + w1 2 w2 2 + w4 2 w2 2

8 -274- from which we conclude that P 2 P 1. Then P 2 is the desired path and the result follows. For n = 6, since w(e 1 ) w(e 5 ), we first show that w(e 2 ) w(e 4 ). Let P be the weighted path obtained from P by exchanging the edges e 1 and e 5, that is, P = u 1 e 5 u 2 e 2 u 3 e 3 u 4 e 4 u 5 e 1 u 6. Then E(P ) E(P ). (5) By Corollary 6, b k (P ) = b k (P \e 1 ) + w(e 1 ) 2 b k 2 (P \u 1 u 2 ) = b k (P \e 1 e 5 ) + w(e 5 ) 2 b k 2 (P \e 1 u 5 u 6 ) + w(e 1 ) 2 b k 2 (P \u 1 u 2 e 5 ) + w(e 1 ) 2 w(e 5 ) 2 b k 4 (P \u 1 u 2 u 5 u 6 ) and b k (P ) = b k (P \e 5 ) + w(e 5 ) 2 b k 2 (P \u 1 u 2 ) = b k (P \e 5 e 1 ) + w(e 1 ) 2 b k 2 (P \e 5 u 5 u 6 ) + w(e 5 ) 2 b k 2 (P \u 1 u 2 e 1 ) + w(e 1 ) 2 w(e 5 ) 2 b k 4 (P \u 1 u 2 u 5 u 6 ) Hence, b k (P ) b k (P ) = ( w(e 5 ) 2 w(e 1 ) 2)[ b k 2 (P \e 1 u 5 u 6 ) b k 2 (P \u 1 u 2 e 5 ) ]. So, we have b 2 (P ) b 2 (P ) = 0 b 4 (P ) b 4 (P ) = ( w(e 5 ) 2 w(e 1 ) 2)[( w(e 2 ) 2 + w(e 3 ) 2) ( w(e 3 ) 2 + w(e 4 ) 2)] b 6 (P ) b 6 (P ) = 0. Since E(P ) E(P ), we have that b 4 (P ) b 4 (P ) 0. From w(e 1 ) w(e 5 ), it follows that w(e 2 ) w(e 4 ). Taking into account Lemmas 8 and 9, we get that the desired path is P 1 = w 1 w 4 w 5 w 3 w 2, or P 2 = w 1 w 5 w 4 w 3 w 2, or P 3 = w 1 w 5 w 3 w 4 w 2. Direct calculation yields that b 2 (P 1 ) = b 2 (P 2 ) = b 2 (P 3 )

9 -275- b 4 (P 1 ) = w 2 1 w w 2 1 w w 2 1 w w 2 4 w w 2 4 w w 2 5 w 2 2 b 4 (P 2 ) = w 2 1 w w 2 1 w w 2 1 w w 2 5 w w 2 5 w w 2 4 w 2 2 b 4 (P 3 ) = w 2 1 w w 2 1 w w 2 1 w w 2 5 w w 2 5 w w 2 3 w 2 2 b 6 (P 1 ) = w 2 1 w 2 5 w 2 2 b 6 (P 2 ) = w 2 1 w 2 4 w 2 2 b 6 (P 3 ) = w 2 1 w 2 3 w 2 2 which implies P 3 P 2 P 1. Then P 3 is the desired path and the result follows. For general case n 7, we cannot solve the problem; but we think that the following conjecture is true. Conjecture 11. For n 3, let W = {w 1, w 2,..., w n 1 } be a sequence of positive numbers such that w 1 w 2 w n 1. Let P = u 1 e 1 u 2 e 2... u n 1 e n 1 u n P(n, W ) be the path having energy Ê(n, W ). Suppose w(e 1) w(e n 1 ). Then for i = 1, 2,..., n 1, w i w(e i ) = otherwise. w n+1 i if either i n 1 2 n 1 and i is odd, or i > and n i is even, 2 it. Though we cannot prove the conjecture, we get the following results to support Theorem 12. For n 3, let W = {x, y,..., y} be a sequence of positive numbers n 2 such that x > y. Then, up to isomorphism, P = x y... y is the unique path having energy Ê(n, W ). n 2 Theorem 12 can be considered as a generalized version of Theorem 11 from Ref. [3], and therefore its proof is omitted. Theorem 13. For n 3, let W = {x, y, z,..., z} be a sequence of positive numbers n 3 such that x y > z. Then, up to isomorphism, P = x z... z y is the unique path having energy Ê(n, W ). n 3

10 -276- Proof. By Theorem 10, the result follows for n 6. Therefore, suppose n 7. Denote by P = u 1 e 1 u 2 e 2 u 3... u n 1 e n 1 u n the path having energy Ê(n, W ). Since there are exactly two edges in P whose weights are not z, we suppose that w(e i ) z and w(e i+t ) z for 1 i n 2, 1 t n 2. Then it is sufficient to show that t = n 2. If t 2 and t n 2, then there exists a pendent edge, say e 1, having weight z. Without loss of generality, suppose w(e i+t ) = x. Then w(e i ) = y. Applying Corollary 6 to the edge e i+t, we have b k (P ) = b k (P \e i+t ) + x 2 b k 2 (P \u i+t u i+t+1 ). Let P be the weighted path obtained from P by exchanging the edges e i and e 1. Then, b k (P ) = b k (P \e i+t ) + x 2 b k 2 (P \u i+t u i+t+1 ). One can show that both P \e i+t and P \e i+t contain a component which is a i+t 2 weighted path belonging to P(i+t, W 1 ), where W 1 = {y, z,..., z}. Applying Theorem 12, we have P \e i+t P \e i+t. Similarly, we have P \u i+t u i+t+1 P \u i+t u i+t+1. Consequently, P P, which yields a contradiction. If t = 1, then there exists a pendent edge, say e 1, having weight z. Without loss of generality, suppose w(e i+1 ) = x. Then w(e i ) = y. Applying Corollary 6 to the edge e i+1, we have b k (P ) = b k (P \e i+1 ) + x 2 b k 2 (P \u i+1 u i+2 ). Let P be the weighted path obtained from P by exchanging the edges e i and e 1. Then, b k (P ) = b k (P \e i+1 ) + x 2 b k 2 (P \u i+1 u i+2 ). One can show that P \e i+1 P \e i+1, and both P \u i+1 u i+2 and P \u i+1 u i+2 contain a component, denoted respectively by P1 and P1, having i vertices. However, i 1 i 2 P1 = z... z and P1 P(i, W 1 ), where W 1 = {y, z,..., z}. Hence, P1 P1. Consequently, P P, which also yields a contradiction. The result thus follows.

11 -277- n 2 Theorem 14. For n 3, let W n 1 = { x,..., x, y} be a sequence of positive numbers such that x > y. Then, up to isomorphism, ˆP n 3 n = xy x... x is the unique path having energy Ê(n, W n 1). Proof. We prove Theorem 14 by induction on n. From Theorem 10, the result follows for n 6. Suppose that the result is true for smaller n. For n 7, denote the path having energy Ê(n, W ) by P = u 1 e 1 u 2 e 2 u 3... u n 1 e n 1 u n. By Lemma 8, w(e 1 ) = w(e n 1 ) = x. Since P has exactly one edge having weight y, we suppose that w(e n 2 ) = x. Applying Corollary 6, we have b k (P ) = b k (P \e n 1 ) + x 2 b k 2 (P \u n 1 u n 2 ). Then, P \e n 1 P(n, W n 2 ) and P \u n 1 u n 2 P(n, W n 3 ). By the induction assumption, P \e n 1 ˆP n 1 with equality holding if and only if P \e n 1 = ˆP n 1. In addition, P \u n 1 u n 2 ˆP n 2 with equality holding if and only if P \u n 1 u n 2 = ˆP n 2. Hence, the result follows. References [1] J. Aihara, General rules for constructing Hückel molecular orbital characteristic polynomials, J. Am. Chem. Soc. 98 (1976) [2] S. Akbari, E. Ghorbani, M. R. Oboudi, Edge addition, singular values and energy of graphs and matrices, Lin. Algebra Appl. 430 (2009) [3] R. A. Brualdi, J. Y. Shao, S. C. Gong, C. Q. Xu, G. H. Xu, On the extremal energy of integral weighted trees, Lin. Multilin. Algebra 60 (2012) [4] C. A. Coulson, On the calculation of the energy in unsaturated hydrocarbon molecules, Proc. Cambridge Phil. Soc. 36 (1940) [5] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs Theory and Application, Academic Press, New York, [6] D. Cvetković, P. Rowlinson, S. Simić, An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, [7] S. C. Gong, G. H. Xu, The characteristic polynomial and the matchings polynomial of a weighted oriented graph, Lin. Algebra Appl. 436 (2012)

12 -278- [8] A. Graovac, I. Gutman, N. Trinajstić, Topological Approach to the Chemistry of Conjugated Molecules, Springer, Berlin, [9] A. Graovac, I. Gutman, N. Trinajstić, T. Živković, Graph theory and molecular orbitals. Application of Sachs theorem, Theor. Chim. Acta 26 (1972) [10] I. Gutman, Acyclic systems with extremal Hückel π-electron energy, Theor. Chim. Acta 45 (1977) [11] I. Gutman, Generalizations of a recurrence relation for the characteristic polynomials of trees, Publ. Inst. Math. (Beograd) 21 (1977) [12] I. Gutman, The energy of a graph, Ber. Math.-Statist. Sekt. Forschungsz. Graz. 103 (1978) [13] I. Gutman, Partial ordering of forests according to their characteristic polynomials, in: A. Hajnal, V. T. Sós (Eds.), Combinatorics, North Holland, Amsterdam, 1978, pp [14] I. Gutman, Rectifying a misbelief: Frank Harary s role in the discovery of the coefficient theorem in chemical graph theory, J. Math. Chem. 16 (1994) [15] I. Gutman, The energy of a graph: Old and new results, in: A. Betten, A. Kohnert, R. Lau, A. Wassermann (Eds.), Algebraic Combinatorics and Applications, Springer, Berlin, 2001, pp [16] I. Gutman, M. Mateljević, Note on the Coulson integral formula, J. Math. Chem. 39 (2006) [17] I. Gutman, J. Y. Shao, The energy change of weighted graphs, Lin. Algebra Appl. 435 (2011) [18] I. Gutman, F. Zhang, On a quasiordering of bipartite graphs, Publ. Inst. Math. (Beograd) 40 (1986) [19] R. B. Mallion, A. J. Schwenk, N. Trinajstić, A graphical study of heteroconjugated molecules, Croat. Chem. Acta 46 (1974) [20] R. B. Mallion, N. Trinajstić, A. J. Schwenk, Graph theory in chemistry Generalization of Sachs formula, Z. Naturforsch. 29a (1974) [21] J. Y. Shao, F. Gong, Z. B. Du, The extremal energies of weighted trees and forests with fixed total weight sum, MATCH Commun. Math. Comput. Chem. 66 (2011) [22] F. Zhang, Two theorems of comparison of bipartite graphs by their energy, Kexue Tongbao 28 (1983) [23] F. Zhang, Z. Lai, Three theorems of comparison of trees by their energy, Sci. Explor. 3 (1983)

### Tricyclic biregular graphs whose energy exceeds the number of vertices

MATHEMATICAL COMMUNICATIONS 213 Math. Commun., Vol. 15, No. 1, pp. 213-222 (2010) Tricyclic biregular graphs whose energy exceeds the number of vertices Snježana Majstorović 1,, Ivan Gutman 2 and Antoaneta

### 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.,

### 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

### 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

### 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

### An inequality for the group chromatic number of a graph

An inequality for the group chromatic number of a graph Hong-Jian Lai 1, Xiangwen Li 2 and Gexin Yu 3 1 Department of Mathematics, West Virginia University Morgantown, WV 26505 USA 2 Department of Mathematics

### 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,

### 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

### 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

### 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

### An inequality for the group chromatic number of a graph

Discrete Mathematics 307 (2007) 3076 3080 www.elsevier.com/locate/disc Note An inequality for the group chromatic number of a graph Hong-Jian Lai a, Xiangwen Li b,,1, Gexin Yu c a Department of Mathematics,

### Cycles in a Graph Whose Lengths Differ by One or Two

Cycles in a Graph Whose Lengths Differ by One or Two J. A. Bondy 1 and A. Vince 2 1 LABORATOIRE DE MATHÉMATIQUES DISCRÉTES UNIVERSITÉ CLAUDE-BERNARD LYON 1 69622 VILLEURBANNE, FRANCE 2 DEPARTMENT OF MATHEMATICS

### All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

### 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

### 22 Gutman and Cioslowski whereas for any bipartite graph X i = X n i+1, i =1; 2;...;n. Since hexagonal systems are bipartite graphs, we immediately de

PUBLICATIONS DE L'INSTITUT MATHÉMATIQUE Nouvelle série tome 42 (56), 1987, pp. 21 27 BOUNDS FOR THE NUMBER OF PERFECT MATCHINGS IN HEXAGONAL SYSTEMS Ivan Gutman and Jerzy Cioslowski Abstract. Upper bounds

### 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

### A Turán Type Problem Concerning the Powers of the Degrees of a Graph

A Turán Type Problem Concerning the Powers of the Degrees of a Graph Yair Caro and Raphael Yuster Department of Mathematics University of Haifa-ORANIM, Tivon 36006, Israel. AMS Subject Classification:

### COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

### Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

### ENERGY OF COMPLEMENT OF STARS

J Indones Math Soc Vol 19, No 1 (2013), pp 15 21 ENERGY OF COMPLEMENT OF STARS P R Hampiholi 1, H B Walikar 2, and B S Durgi 3 1 Department of Master of Computer Applications, Gogte Institute of Technology,

### 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

### 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

### A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

### 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

### A characterization of trace zero symmetric nonnegative 5x5 matrices

A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the

### most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, The University of Newcastle Callaghan, NSW 2308, Australia University of West Bohemia

Complete catalogue of graphs of maimum degree 3 and defect at most 4 Mirka Miller 1,2, Guillermo Pineda-Villavicencio 3, 1 School of Electrical Engineering and Computer Science The University of Newcastle

### 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

### 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

### Non-Separable Detachments of Graphs

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2001-12. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### 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

### 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

### 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:

### 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

### ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER

Séminaire Lotharingien de Combinatoire 53 (2006), Article B53g ON DEGREES IN THE HASSE DIAGRAM OF THE STRONG BRUHAT ORDER RON M. ADIN AND YUVAL ROICHMAN Abstract. For a permutation π in the symmetric group

### 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

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

### 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

### 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

### 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

### The chromatic spectrum of mixed hypergraphs

The chromatic spectrum of mixed hypergraphs Tao Jiang, Dhruv Mubayi, Zsolt Tuza, Vitaly Voloshin, Douglas B. West March 30, 2003 Abstract A mixed hypergraph is a triple H = (X, C, D), where X is the vertex

### Suk-Geun Hwang and Jin-Woo Park

Bull. Korean Math. Soc. 43 (2006), No. 3, pp. 471 478 A NOTE ON PARTIAL SIGN-SOLVABILITY Suk-Geun Hwang and Jin-Woo Park Abstract. In this paper we prove that if Ax = b is a partial signsolvable linear

### 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

### 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

### On end degrees and infinite cycles in locally finite graphs

On end degrees and infinite cycles in locally finite graphs Henning Bruhn Maya Stein Abstract We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel

### 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

### 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

### SOME RESULTS ON THE DRAZIN INVERSE OF A MODIFIED MATRIX WITH NEW CONDITIONS

International Journal of Analysis and Applications ISSN 2291-8639 Volume 5, Number 2 (2014, 191-197 http://www.etamaths.com SOME RESULTS ON THE DRAZIN INVERSE OF A MODIFIED MATRIX WITH NEW CONDITIONS ABDUL

### Generalized Induced Factor Problems

Egerváry Research Group on Combinatorial Optimization Technical reports TR-2002-07. Published by the Egrerváry Research Group, Pázmány P. sétány 1/C, H 1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres.

### 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

### On the Modes of the Independence Polynomial of the Centipede

47 6 Journal of Integer Sequences, Vol. 5 0), Article.5. On the Modes of the Independence Polynomial of the Centipede Moussa Benoumhani Department of Mathematics Faculty of Sciences Al-Imam University

### 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

### Graph Classification and Easy Reliability Polynomials

Mathematical Assoc. of America American Mathematical Monthly 121:1 November 18, 2014 1:11 a.m. AMM.tex page 1 Graph Classification and Easy Reliability Polynomials Pablo Romero and Gerardo Rubino 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

### 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

### M-Degrees of Quadrangle-Free Planar Graphs

M-Degrees of Quadrangle-Free Planar Graphs Oleg V. Borodin, 1 Alexandr V. Kostochka, 1,2 Naeem N. Sheikh, 2 and Gexin Yu 3 1 SOBOLEV INSTITUTE OF MATHEMATICS NOVOSIBIRSK 630090, RUSSIA E-mail: brdnoleg@math.nsc.ru

### 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,

### 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

### 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

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### MITES Physics III Summer Introduction 1. 3 Π = Product 2. 4 Proofs by Induction 3. 5 Problems 5

MITES Physics III Summer 010 Sums Products and Proofs Contents 1 Introduction 1 Sum 1 3 Π Product 4 Proofs by Induction 3 5 Problems 5 1 Introduction These notes will introduce two topics: A notation which

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### 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

### ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL

ADSS, Volume 3, Number 3, 2013, Pages 45-56 2013 Aditi International ON THE COEFFICIENTS OF THE LINKING POLYNOMIAL KOKO KALAMBAY KAYIBI Abstract Let i j T( M; = tijx y be the Tutte polynomial of the matroid

### Split Nonthreshold Laplacian Integral Graphs

Split Nonthreshold Laplacian Integral Graphs Stephen Kirkland University of Regina, Canada kirkland@math.uregina.ca Maria Aguieiras Alvarez de Freitas Federal University of Rio de Janeiro, Brazil maguieiras@im.ufrj.br

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### 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

### 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.

### Analysis of Algorithms, I

Analysis of Algorithms, I CSOR W4231.002 Eleni Drinea Computer Science Department Columbia University Thursday, February 26, 2015 Outline 1 Recap 2 Representing graphs 3 Breadth-first search (BFS) 4 Applications

### Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

### arxiv: v2 [math.co] 30 Nov 2015

PLANAR GRAPH IS ON FIRE PRZEMYSŁAW GORDINOWICZ arxiv:1311.1158v [math.co] 30 Nov 015 Abstract. Let G be any connected graph on n vertices, n. Let k be any positive integer. Suppose that a fire breaks out

### Outline 2.1 Graph Isomorphism 2.2 Automorphisms and Symmetry 2.3 Subgraphs, part 1

GRAPH THEORY LECTURE STRUCTURE AND REPRESENTATION PART A Abstract. Chapter focuses on the question of when two graphs are to be regarded as the same, on symmetries, and on subgraphs.. discusses the concept

### n 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1

. Calculate the sum of the series Answer: 3 4. n 2 + 4n + 3. The answer in decimal form (for the Blitz):, 75. Solution. n 2 + 4n + 3 = (n + )(n + 3) = (n + 3) (n + ) = 2 (n + )(n + 3) ( 2 n + ) = m ( n

### RAMSEY-TYPE NUMBERS FOR DEGREE SEQUENCES

RAMSEY-TYPE NUMBERS FOR DEGREE SEQUENCES ARTHUR BUSCH UNIVERSITY OF DAYTON DAYTON, OH 45469 MICHAEL FERRARA, MICHAEL JACOBSON 1 UNIVERSITY OF COLORADO DENVER DENVER, CO 80217 STEPHEN G. HARTKE 2 UNIVERSITY

### 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

### Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

### Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph

FPSAC 2009 DMTCS proc (subm), by the authors, 1 10 Determinants in the Kronecker product of matrices: The incidence matrix of a complete graph Christopher R H Hanusa 1 and Thomas Zaslavsky 2 1 Department

### Math 115A HW4 Solutions University of California, Los Angeles. 5 2i 6 + 4i. (5 2i)7i (6 + 4i)( 3 + i) = 35i + 14 ( 22 6i) = 36 + 41i.

Math 5A HW4 Solutions September 5, 202 University of California, Los Angeles Problem 4..3b Calculate the determinant, 5 2i 6 + 4i 3 + i 7i Solution: The textbook s instructions give us, (5 2i)7i (6 + 4i)(

### 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

### 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

### The positive minimum degree game on sparse graphs

The positive minimum degree game on sparse graphs József Balogh Department of Mathematical Sciences University of Illinois, USA jobal@math.uiuc.edu András Pluhár Department of Computer Science University

### DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH

DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,

### MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

### Computer Science Department. Technion - IIT, Haifa, Israel. Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

- 1 - THREE TREE-PATHS Avram Zehavi Alon Itai Computer Science Department Technion - IIT, Haifa, Israel Abstract Itai and Rodeh [IR] have proved that for any 2-connected graph G and any vertex s G there

### ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS

ZERO-DIVISOR GRAPHS OF POLYNOMIALS AND POWER SERIES OVER COMMUTATIVE RINGS M. AXTELL, J. COYKENDALL, AND J. STICKLES Abstract. We recall several results of zero divisor graphs of commutative rings. We

### Explicit inverses of some tridiagonal matrices

Linear Algebra and its Applications 325 (2001) 7 21 wwwelseviercom/locate/laa Explicit inverses of some tridiagonal matrices CM da Fonseca, J Petronilho Depto de Matematica, Faculdade de Ciencias e Technologia,

### Mathematical Induction. Lecture 10-11

Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach

### AN ITERATIVE METHOD FOR THE HERMITIAN-GENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT

Bulletin of the Iranian Mathematical Society Vol. 39 No. 6 (2013), pp 129-1260. AN ITERATIVE METHOD FOR THE HERMITIAN-GENERALIZED HAMILTONIAN SOLUTIONS TO THE INVERSE PROBLEM AX = B WITH A SUBMATRIX CONSTRAINT

### Smooth functions statistics

Smooth functions statistics V. I. rnold To describe the topological structure of a real smooth function one associates to it the graph, formed by the topological variety, whose points are the connected

### Cycle transversals in bounded degree graphs

Electronic Notes in Discrete Mathematics 35 (2009) 189 195 www.elsevier.com/locate/endm Cycle transversals in bounded degree graphs M. Groshaus a,2,3 P. Hell b,3 S. Klein c,1,3 L. T. Nogueira d,1,3 F.

### A Sublinear Bipartiteness Tester for Bounded Degree Graphs

A Sublinear Bipartiteness Tester for Bounded Degree Graphs Oded Goldreich Dana Ron February 5, 1998 Abstract We present a sublinear-time algorithm for testing whether a bounded degree graph is bipartite

### The Characteristic Polynomial

Physics 116A Winter 2011 The Characteristic Polynomial 1 Coefficients of the characteristic polynomial Consider the eigenvalue problem for an n n matrix A, A v = λ v, v 0 (1) The solution to this problem

### IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction

IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible

### Discrete Mathematics Problems

Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................

### Just the Factors, Ma am

1 Introduction Just the Factors, Ma am 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

### Sample Induction Proofs

Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given

### 1.1 Logical Form and Logical Equivalence 1

Contents Chapter I The Logic of Compound Statements 1.1 Logical Form and Logical Equivalence 1 Identifying logical form; Statements; Logical connectives: not, and, and or; Translation to and from symbolic