On the log-convexity of combinatorial sequences

We give new sufficient conditions for a sequence of polynomials to have only real zeros based on the method of interlacing zeros. As applications we derive several well-known facts, including the reality of zeros of orthogonal polynomials, matching polynomials, Narayana polynomials and Eulerian polynomials. We also settle certain conjectures of Stahl on genus polynomials by proving them for certain classes of graphs, while showing that they are false in general.     

Fractional chromatic numbers of tensor products of three graphs

The tensor product $(G_1,G_2,G_3)$ of graphs $G_1$, $G_2$ and $G_3$ is defined by $V(G_1,G_2,G_3)=V\left(G_1\right)\times V\left(G_2\right)\times V\left(G_3\right)$and $E(G_1,G_2,G_3)=\left\{\left((u_1,u_2,u_3),(v_1,v_2,v_3)\right):\left|\{i:(u_i,v_i)\in E\left(G_i\right)\}\right|\ge 2\right\}.$Let ${\chi }_f\left(G\right)$ be the fractional chromatic number of a graph $G$. In this paper, we prove that if one of the three graphs $G_1$, $G_2$ and $G_3$ is a circular clique, ${\chi }_f(G_1,G_2,G_3)=min\{{\chi }_f\left(G_1\right){\chi }_f\left(G_2\right),{\chi }_f\left(G_1\right){\chi }_f\left(G_3\right),{\chi }_f\left(G_2\right){\chi }_f\left(G_3\right)\}.$     

Estimates for the real and imaginary parts of the eigenvalues of matrices and applications

This paper is a continuation of our recent work on the localization of the eigenvalues of matrices. We give new bounds for the real and imaginary parts of the eigenvalues of matrices. Applications to the localization of the zeros of polynomials are also given.     

Enumerating several aspects of non-decreasing Dyck paths

A Dyck path is non-decreasing if the $y$-coordinates of its valleys form a non-decreasing sequence. In this paper we give enumerative results and some statistics of several aspects of non-decreasing Dyck paths. We give the number of pyramids at a fixed level that the paths of a given length have, count the number of primitive paths, count how many of the non-primitive paths can be expressed as a product of primitive paths, and count the number of paths of a given height and a given length. We present and prove our results using combinatorial arguments, generating functions (using the symbolic method) and parameterize the results studied here using the Riordan arrays. We use known bijections to connect direct column-convex polyominoes, Elena trees, and non-decreasing Dyck paths.     

The Expected Number of Local Maxima of a Random Algebraic Polynomial

In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial , where a _j (j=0, 1,...,n–1) are independent, normally distributed random variables with mean and variance one. It is shown that for nonzero , the expected number of maxima is asymptotic to log n, when n is large.     

An Euler‐genus approach to the calculation of the crosscap‐number polynomial

In 1994, J. Chen, J. Gross, and R. Rieper demonstrated how to use the rank of Mohar's overlap matrix to calculate the crosscap‐number distribution, that is, the distribution of the embeddings of a graph in the nonorientable surfaces. That has ever since been by far the most frequent way that these distributions have been calculated. This article introduces a way to calculate the Euler‐genus polynomial of a graph, which combines the orientable and the nonorientable embeddings, without using the overlap matrix. The crosscap‐number polynomial for the nonorientable embeddings is then easily calculated from the Euler‐genus polynomial and the genus polynomial.