On minimum rank and zero forcing sets of a graph

For a graph G on n vertices and a field F, the minimum rank of G over F, written as mr^F(G), is the smallest possible rank over all n×n symmetric matrices over F whose (i,j)th entry (for ) is nonzero whenever ij is an edge in G and is zero otherwise. The maximum nullity of G over F is M^F(G)=n-mr^F(G). The minimum rank problem of a graph G is to determine mr^F(G) (or equivalently, M^F(G)). This problem has received considerable attention over the years. In [F. Barioli, W. Barrett, S. Butler, S.M. Cioab?, D. Cvetkovi?, S.M. Fallat, C. Godsil, W. Haemers, L. Hogben, R. Mikkelson, S. Narayan, O. Pryporova, I. Sciriha, W. So, D. Stevanovi?, H. van der Holst, K.V. Meulen, A.W. Wehe, AIM Minimum Rank-Special Graphs Work Group, Zero forcing sets and the minimum rank of graphs, Linear Algebra Appl. 428 (2008) 1628-1648], a new graph parameter Z(G), the zero forcing number, was introduced to bound M^F(G) from above. The authors posted an attractive question: What is the class of graphs G for which Z(G)=M^F(G) for some field F? This paper focuses on exploring the above question.     

On the spectral spread of bicyclic graphs with given girth

The spectral spread of a graph is defined to be the difference between the largest and the least eigenvalue of the adjacency matrix of the graph. A graph G is said to be bicyclic, if G is connected and |E(G)| = |V(G)|+ 1. Let B(n, g) be the set of bicyclic graphs on n vertices with girth g. In this paper some properties about the least eigenvalues of graphs are given, by which the unique graph with maximal spectral spread in B(n, g) is determined.     

Expected rank and randomness in rooted graphs

For a rooted graph G, let EV_b(G;p) be the expected number of vertices reachable from the root when each edge has an independent probability p of operating successfully. We determine the expected value of EV_b(G;p) for random trees, and include a connection to unrooted trees. We also consider rooted digraphs, computing the expected value of a random orientation of a rooted graph G in terms of EV_b(G;p). We consider optimal location of the root vertex for the class of grid graphs, and we also briefly discuss a polynomial that incorporates vertex failure.     

A note on universally optimal matrices and field independence of the minimum rank of a graph

For a simple graph G on n vertices, the minimum rank of G over a field F, written as mr^F(G), is defined to be the smallest possible rank among all n×n symmetric matrices over F whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. A symmetric integer matrix A such that every off-diagonal entry is 0, 1, or -1 is called a universally optimal matrix if, for all fields F, the rank of A over F is the minimum rank of the graph of A over F. Recently, Dealba et al. [L.M. Dealba, J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen, Universally optimal matrices and field independence of the minimum rank of a graph, Electron. J. Linear Algebra 18 (2009) 403-419] initiated the study of universally optimal matrices and established field independence or dependence of minimum rank for some families of graphs. In the present paper, more results on universally optimal matrices and field independence or dependence of the minimum rank of a graph are presented, and some results of Dealba et al. [5] are improved.     

On the minimum semidefinite rank of a simple graph

The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.     

The rank of random graphs

We show that almost surely the rank of the adjacency matrix of the Erd?s‐Rényi random graph G(n,p) equals the number of nonisolated vertices for any c ln n/n ≤ p ≤ 1/2, where c is an arbitrary positive constant larger than 1/2. In particular, the adjacency matrix of the giant component (a.s.) has full rank in this range. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The arithmetical rank of the edge ideals of cactus graphs

We prove that, for the edge ideal of a cactus graph, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes. The bound is sharp, but in many cases, it can be improved. Moreover, we show that the edge ideal of a Cohen–Macaulay graph that contains exactly one cycle or is chordal or has no cycles of length 4 and 5 is a set-theoretic complete intersection.     

秩$6$的无三角形的图

The rank of a graph G is defined to be the rank of its adjacency matrix A（G）. In this paper we characterize all connected triangle-free graphs with rank 6.     

On mixed block graphs

A mixed complete graph is obtained from a directed cycle of length at least three by adding all the possible arcs between any non-adjacent vertices of the underlying cycle. A mixed block graph is a strongly connected directed graph whose blocks are mixed complete graphs. In this paper, we give the inverse of the distance matrix of the mixed block graph.