Characteristic polynomials of symmetric graphs

Characteristic polynomials of real symmetric random matrices

It is shown that the correlation functions of the random variables det(λ−X), in which X is a real symmetric N × N random matrix, exhibit universal local statistics in the large N limit. The derivation relies on an exact dual representation of the problem: the k-point functions are expressed in terms of finite integrals over (quaternionic) k × k matrices. However the control of the Dyson limit, ...

Small-Span Characteristic Polynomials of Integer Symmetric Matrices

Let f(x) ∈ Z[x] be a totally real polynomial with roots α1 ≤ . . . ≤ αd. The span of f(x) is defined to be αd − α1. Monic irreducible f(x) of span less than 4 are special. In this paper we give a complete classification of those small-span polynomials which arise as characteristic polynomials of integer symmetric matrices. As one application, we find some low-degree polynomials that do not aris...

Tutte Polynomials of Flower Graphs

Characteristic Polynomials of Skew-Adjacency Matrices of Oriented Graphs

An oriented graph Gσ is a simple undirected graph G with an orientation, which assigns to each edge a direction so that Gσ becomes a directed graph. G is called the underlying graph of Gσ and we denote by S(Gσ) the skew-adjacency matrix of Gσ and its spectrum Sp(Gσ) is called the skew-spectrum of Gσ. In this paper, the coefficients of the characteristic polynomial of the skew-adjacency matrix S...

On the log-concavity of Laplacian characteristic polynomials of graphs

Let G be a graph and L(G) be the Laplacian matrix of G. In this article, we first point out that the sequence of the moduli of Laplacian coefficients of G is log-concave and hence unimodal. Using this fact, we provide an upper bound for the partial sums of the Laplacian eigenvalues of G, based on coefficients of its Laplacian characteristic polynomial. We then obtain some lower bounds on the al...