We consider the cop-throttling number of a graph G for the game of Cops and Robbers, which is defined to be the minimum of (k + capt k (G)), where k is the number of cops and capt k (G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games. We provide some tools for bounding the copthrottling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are O( √ n).
In a graph theory setting, Kemeny’s constant is a graph parameter which measures a weighted average of the mean first passage times in a random walk on the vertices of the graph. In one sense, Kemeny’s constant is a measure of how well the graph is ‘connected’. An explicit computation for this parameter is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length, as well as for two cliques joined by two paths of different length. In each case, Kemeny’s constant is shown to be O(n3), which is the largest possible order of Kemeny’s constant for a graph on n vertices. The approach used is based on interesting techniques in spectral graph theory and includes a generalization of using twin subgraphs to find the spectrum of a graph.
The computation of eigenvalues and eigenvectors of matrix polynomials is an important, but difficult, problem. The standard approach to solve this problem is to use linearizations, which are matrix polynomials of degree 1 that share the eigenvalues of P (λ).Hermitian matrix polynomials and their real eigenvalues are of particular interest in applications. Attached to these eigenvalues is a set of signs called the sign characteristic. From both a theoretical and a practical point of view, it is important to be able to recover the sign characteristic of a Hermitian linearization of P (λ) from the sign characteristic of P (λ).In this paper, for a Hermitian matrix polynomial P (λ) with nonsingular leading coefficient, we describe, in terms of the sign characteristic of P (λ), the sign characteristic of the Hermitian linearizations in the vector space DL(P ) (Mackey, Mackey, Mehl and Merhmann, 2006). In particular, we identify the Hermitian linearizations in DL(P ) that preserve the sign characteristic of P (λ). We also provide a description of the sign characteristic of the Hermitian linearizations of P (λ) in the family of generalized Fiedler pencils with repetition (Bueno, Dopico, Furtado and Rychnovsky, 2015).
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