2 Eigenvalues of graphs 5 2.1 Matrices associated with graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The largest eigenvalue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Adjacency matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.3...

Question 1 First, we have that λ − λ − 2 = (λ + 1)(λ − 2). Therefore, it is true that the eigenvalues of the corresponding adjacency matrix come in pairs of additive inverse. However, notice that ± √ −1 are eigenvalues, and therefore the associated matrix cannot be symmetric (there would only be real eigenvalues). We conclude that no undirected graph can have the above characteristic polynomial...

The eigenvalues of a digraph are the eigenvalues of its adjacency matrix. The sum of the absolute values of the real part of the eigenvalues is called the energy of the digraph. The extremal energy of bicyclic digraphs with vertex-disjoint directed cycles is known. In this paper, we consider a class of bicyclic digraphs with exactly two linear subdigraphs of equal length. We find the minimal an...

We derive the joint limiting distribution for the largest eigenvalues of the adjacency matrix for stochastic blockmodel graphs when the number of vertices tends to infinity. We show that, in the limit, these eigenvalues are jointly multivariate normal with bounded covariances. Our result extends the classical result of Füredi and Komlós on the fluctuation of the largest eigenvalue for Erdős-Rén...

let $g$ be a simple graph‎, ‎and $g^{sigma}$‎ ‎be an oriented graph of $g$ with the orientation ‎$sigma$ and skew-adjacency matrix $s(g^{sigma})$‎. ‎the $k-$th skew spectral‎ ‎moment of $g^{sigma}$‎, ‎denoted by‎ ‎$t_k(g^{sigma})$‎, ‎is defined as $sum_{i=1}^{n}( ‎‎‎lambda_{i})^{k}$‎, ‎where $lambda_{1}‎, ‎lambda_{2},cdots‎, ‎lambda_{n}$ are the eigenvalues of $g^{sigma}$‎. ‎suppose‎ ‎$g^{sigma...

is the diagonal matrix of vertex degrees of G and A(G) is the adjacency matrix ofG. The eigenvalues of L(G) are called the Laplacian eigenvalues and denoted by λ1 ≥ λ2 ≥ · · · ≥ λn = 0. It is well known that λ1 ≤ n. We denote the number of spanning trees (also known as complexity) of G by κ(G). The following formula in terms of the Laplacian eigenvalues of G is well known (see, for example, [2]...

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. In this study the upper bounds for the spectral radius of weighted graphs, which edge weights are positive definite matrices, are compared. Mathematics Subject Classification: 05C50

A multicone graph is defined to be join of a clique and a regular graph. A graph $ G $ is cospectral with graph $ H $ if their adjacency matrices have the same eigenvalues. A graph $ G $ is said to be determined by its spectrum or DS for short, if for any graph $ H $ with $ Spec(G)=Spec(H)$, we conclude that $ G $ is isomorphic to $ H $. In this paper, we present new classes of multicone graphs...

A signed adjacency matrix is a {−1, 0, 1}-matrix A obtained from the adjacency matrix A of a simple graph G by symmetrically replacing some of the 1’s of A by −1’s. Bilu and Linial have conjectured that if G is k-regular, then some A has spectral radius ρ(A) ≤ 2 √ k − 1. If their conjecture were true then, for each fixed k > 2, it would immediately guarantee the existence of infinite families o...