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On Unimodular Graphs View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 421 (2007) 3–15 www.elsevier.com/locate/laa On unimodular graphs S. Akbari a, S.J. Kirkland b,∗ a Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran b Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 Received 21 October 2005; accepted 18 October 2006 Submitted by R. Bhatia Abstract We study graphs whose adjacency matrices have determinant equal to 1 or −1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also character- ized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address the problem of when that inverse is diagonally similar to a non-negative matrix. Special attention is paid to the case that such a graph is unicyclic. © 2006 Elsevier Inc. All rights reserved. AMS classification: 05C05; 05C50; 15A09; 15A15 Keywords: Determinant; Minor; Forest; Unicyclic graph; Unimodular graph 1. Introduction and preliminaries Let G be a graph with adjacency matrix A. Evidently if A is invertible, then its inverse is an integral matrix if and only if det(A) =±1. In a similar vein, if A is singular, there is a full rank submatrix, say B, so that the rows and columns of A can be permuted and partitioned as BC A = , CT D where A and B have equal rank (see [9, p. 179]). If det(B) =±1, then the reduced row echelon form for A is the integral matrix ∗ Corresponding author. Tel.: +1 306 585 4352; fax: +1 306 585 4020. E-mail addresses: [email protected] (S. Akbari), [email protected] (S.J. Kirkland). 0024-3795/$ - see front matter ( 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2006.10.017 4 S. Akbari, S.J. Kirkland / Linear Algebra and its Applications 421 (2007) 3–15 IB−1C 00 (see [1] for more on the reduced row echelon form for certain classes of graphs). These observations prompt our interest in graphs whose adjacency matrices have determinant 1 or −1, and those graphs are the subject of this paper. Throughout this paper, we consider simple graphs with no loops or multiple edges, and we will use standard terminology and ideas from both graph theory and matrix theory. We refer the reader to [3] for background on the former and to [11] for background on the latter. For convenience, we recall a few key notions that will be used in the paper. We denote a path and a cycle on k vertices, by Pk and Ck, respectively. A forest is a graph with no cycles, and a connected forest is known as a tree. A graph is called unicyclic if it is connected and contains exactly one cycle. A matching of a graph G is a subgraph of G which is 1-regular. The number of edges in a matching is called its size.Amaximum matching is a matching with largest size. A perfect matching of G is a spanning matching of G. Clearly, if G has a perfect matching, then the number of vertices of G is even. A vertex is a pendant vertex if it has degree 1; a pendant edge is an edge that is incident with a pendant vertex. Let G be a graph with vertex set {v1,...,vn}. The adjacency matrix of G is the n × n matrix A such that Aij is1ifvi and vj are adjacent and is 0 otherwise. We will occasionally use det(G) to denote the determinant of the adjacency matrix for G. Remark 1. Let G be a bipartite graph with two parts of sizes m and n. Let B A = 0 BT 0 be the adjacency matrix of G.Ifm/= n, then B does not have full rank and so det(A) = 0. If m = n, then we have det(A) = (−1)n(det(B))2. Clearly, every perfect matching of G corresponds to a transversal in B with all cells 1, and conversely. Consequently, if G has no perfect matching, then det(G) = 0. If det(G) = 1, then the number of perfect matchings is odd. Also if G has exactly one perfect matching, then det(G) = (−1)n. Recall that a unimodular matrix is a square matrix with determinant 1 or −1, while a rectangular matrix is called totally unimodular if each of its minors is −1, 0 or 1. Similarly, a square matrix is called principally unimodular if each of its principal minors is −1, 0, or 1. Consequently, we will say that a graph is unimodular, totally unimodular or principally unimodular, according as its adjacency matrix is unimodular, totally unimodular or principally unimodular, respectively. Similarly, we will say that a graph is unital if its adjacency matrix has determinant 0, 1or−1. Evidently, the class of unital graphs properly includes the class of unimodular graphs. A signed bipartite graph is a bipartite graph in which each edge is given a weight of −1or +1. The weight of a cycle in a signed bipartite graph is the sum of the weights of its edges. A signed bipartite graph is called restricted unimodular if the weight of any cycle in G is divisible by 4. There is a natural correspondence between a signed bipartite graph G and its {−1, 0, 1} adjacency matrix A, where the rows and columns of A are indexed by the vertices of G, with Aij =−1, 0, 1, according as the edge between vertices vi and vj has weight −1, is absent, or has weight +1, respectively. The following result is due to Commoner [5]. Theorem A. A signed bipartite graph that is restricted unimodular has a totally unimodular adjacency matrix. S. Akbari, S.J. Kirkland / Linear Algebra and its Applications 421 (2007) 3–15 5 Fig. 1. G is totally unimodular but not restricted unimodular. Consider the graph G shown in Fig. 1. Evidently, G is bipartite, but if each edge weight is 1, then it is not restricted unimodular. A straightforward check shows that G is totally unimodular, so that the converse of Theorem A fails. An elementary graph is a graph G, each component of which is either a single edge or a cycle; let s(G) denote the number of components that are cycles. The following theorem of Harary (see [2, p. 44, 50]) uses elementary graphs to compute the determinant. Theorem B. For any graph G, we have |E()|−s() s() det(G) = (−1) 2 , where the summation is over all spanning elementary subgraphs of G. Equivalently, we have r s det(G) = f(r,s)(−1) 2 , where f(r,s)is the number of spanning elementary subgraphs with s cycles and r + s edges. Corollary 1. Let F be a forest on n vertices. If F does not have a perfect matching, then det(F ) = 0. If F has a perfect matching, then det(F ) = (−1)n/2. n− Corollary 2. If n is odd, then det(Cn) = (−1) 12. If n ≡ 0 mod 4, then det(Cn) = 0, while if n ≡ 2 mod 4, then det(Cn) =−4. Remark 2. If a graph G has no perfect matching, then its determinant is an even number. Thus unimodular graphs have an even number of vertices. 2. Unital and unimodular graphs In this section, we explore constructions of and characterizations for unital and unimodular graphs. Theorem 1. Let A be an n by n integral symmetric matrix with even entries on the main diagonal. Then (i) if n ≡ 0, mod 4, then det(A) ≡ 0 or 1, mod 4; (ii) if n ≡ 2 mod 4; then det(A) ≡ 0 or −1 mod 4. Proof. We proceed by induction on n, and note that if n = 2, then the assertion holds. Now suppose that n>2 is even. If all entries of A are even, then det(A) is 0 mod 4. Thus we may assume that x1 y there exists a principal submatrix of A, say P = yx, where x1 and x2 are even and y is odd. 2 −x y PBT We have P −1 = 2 mod 4. Writing, without loss of generality, A = , we have, y −x1 BC from the Schur complement formula for the determinant, det(A) = det(P ) det(C − BP−1BT) = −det(C − BP−1BT) mod 4. It is easily checked that C − BP−1BT is a symmetric integral matrix 6 S. Akbari, S.J. Kirkland / Linear Algebra and its Applications 421 (2007) 3–15 each of whose entries on the diagonal are even; applying the induction hypothesis now yields the desired conclusion. The following is immediate from Remark 2 and Theorem 1. Corollary 3. Let G be a graph of order n. If det(G) = 1, then n is divisible by 4, while if det(G) =−1, then n is congruent to 2 mod 4. The proof of the next lemma is simple and we omit it. Lemma 1. Let G be a graph with pendant vertex v which is adjacent to w. Then det(G) = −det(G \{v,w}). In particular,Gis unimodular (respectively, unital) if and only if G \{v,w} is unimodular (respectively, unital). Theorem 2. Let G be a unicyclic graph. Then det(G) ∈{0, ±1, ±2, ±4}. Proof. By Lemma 1 there is a subgraph of G, say H , whose connected components are either the cycle or are trees and such that |det(G)|=|det(H )|.
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