We will discuss a few basic facts about the distribution of eigenvalues of the adjacency matrix, and some applications. Then we discuss the question of computing the eigenvalues of a symmetric matrix. 1 Eigenvalue distribution Let us consider a d-regular graph G on n vertices. Its adjacency matrix AG is an n× n symmetric matrix, with all of its eigenvalues lying in [−d, d]. How are the eigenval...

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the e...

Consider the random matrix obtained from the adjacency matrix of a random d-regular graph by multiplying every entry by a random sign. The largest eigenvalue converges, after proper scaling, to the Tracy–Widom distribution.

Godsil-McKay switching is an operation on graphs that doesn’t change the spectrum of the adjacency matrix. Usually (but not always) the obtained graph is non-isomorphic with the original graph. We present a straightforward sufficient condition for being isomorphic after switching, and give examples which show that this condition is not necessary. For some graph products we obtain sufficient con...

We provide a useful method for calculating the state vector of a state equation efficiently in a max-plus algebraic system. For a discrete event system whose precedence relationships are represented by a directed acyclic graph, computing the transition matrix, which includes the Kleene star operation of a weighted adjacency matrix, is occasionally the bottleneck. On the other hand, the common o...

There is a connection between the expansion of a graph and the eigengap (or spectral gap) of the normalized adjacency matrix (that is, the gap between the first and second largest eigenvalues). Recall that the largest eigenvalue of the normalized adjacency matrix is 1; denote it by λ1 and denote the second largest eigenvalue by λ2. We will see that a large gap (that is, small λ2) implies good e...

This paper initiates the study of testing properties of directed graphs. In particular, the paper considers the most basic property of directed graphs – acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacency matrix representation, most appropriate for dense graphs, a testing algorithm is developed t...

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain another upper bound which is sharp on the spectral radius of the adjacency matrix and compare with some known upper bounds with the help of some examples of graphs. We also characterize graphs for which the bound is attained.

There is a Paley graph for each prime power q such that q ≡ 1 (mod 4). The vertex set is the field Fq and two vertices x and y are joined by an edge if and only if x − y is a nonzero square of Fq. We compute the Smith normal forms of the adjacency matrix and Laplacian matrix of a Paley graph.

For two graphs $mathrm{G}$ and $mathrm{H}$ with $n$ and $m$ vertices, the corona $mathrm{G}circmathrm{H}$ of $mathrm{G}$ and $mathrm{H}$ is the graph obtained by taking one copy of $mathrm{G}$ and $n$ copies of $mathrm{H}$ and then joining the $i^{th}$ vertex of $mathrm{G}$ to every vertex in the $i^{th}$ copy of $mathrm{H}$. The neighborhood corona $mathrm{G}starmathrm{H}$ of $mathrm{G}$ and $...