The 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, ...

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...

On the Laplacian characteristic polynomials of mixed graphs

Let G be a mixed graph and L(G) be the Laplacian matrix of G. In this paper, the coefficients of the Laplacian characteristic polynomial of G are studied. The first derivative of the characteristic polynomial of L(G) is explicitly expressed by means of Laplacian characteristic polynomials of its edge deleted subgraphs. As a consequence, it is shown that the Laplacian characteristic polynomial o...

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