The rank of the adjacency matrix of a graph over the real field is a natural parameter associated with the graph, and there has been substantial interest in relating this to other graph theoretic parameters. In particular the relationship between the rank and chromatic number of a graph, and the rank and the maximum number of vertices of a graph has been extensively studied. This was at least i...

A graph G is said to be singular if its adjacency matrix is singular; otherwise it is said to be non-singular. In this paper, we introduce a class of graphs called looped-trees, and find the determinant and the nonsingularity of looped-trees. Moreover, we determine the singularity or non-singularity of the complement of a certain class of trees with diameter 5 by using the results for looped-tr...

Recently, Berman and Zhang (Discrete Math. 260 (2003) 27) obtained an upper bound for the bipartite density of cubic graphs in terms of the smallest eigenvalue of an adjacency matrix and attempted to characterize graphs for which the upper bound is attained. Here we do characterize graphs for which the upper bound is attained, and correct a few errors from Berman and Zhang’s paper. c © 2004 Els...

Let A be a (0, 1, ∗)-matrix with main diagonal all 0’s and such that if ai,j = 1 or ∗ then aj,i = ∗ or 0. Underwhat conditions on the row sums, and or column sums, of A is it possible to change the ∗’s to 0’s or 1’s and obtain a tournament matrix (the adjacency matrix of a tournament) with a specified score sequence? We answer this question in the case of regular and nearly regular tournaments....

Let G be a graph on n vertices with spectral radius λ (this is the largest eigenvalue of the adjacency matrix of G). We show that if G does not contain the complete bipartite graph Kt,s as a subgraph, where 2 6 t 6 s, then λ 6 (

Let M be a mixed graph and [Formula: see text] be its Hermitian-adjacency matrix. If we add a Randić weight to every edge and arc in M, then we can get a new weighted Hermitian-adjacency matrix. What are the properties of this new matrix? Motivated by this, we define the Hermitian-Randić matrix [Formula: see text] of a mixed graph M, where [Formula: see text] ([Formula: see text]) if [Formula: ...

We propose an inexact method for the graph Fourier transform of a graph signal, as defined by the signal decomposition over the Jordan subspaces of the graph adjacency matrix. This method projects the signal over the generalized eigenspaces of the adjacency matrix, which accelerates the transform computation over large, sparse, and directed adjacency matrices. The trade-off between execution ti...

Percolation threshold of a network is the critical value such that when nodes or edges are randomly selected with probability below the value, the network is fragmented but when the probability is above the value, a giant component connecting a large portion of the network would emerge. Assessing the percolation threshold of networks has wide applications in network reliability, information spr...

Let Γ denote an undirected, connected, regular graph with vertex set X, adjacency matrix A, and d+1 distinct eigenvalues.

A characterization of the adjacency matrix of the line digraph of a regular digraph is given. Some corollaries are observed.