The rank of Pseudo walk matrices: controllable and recalcitrant pairs

Abstract

A pseudo walk matrix \(\mathbf{W}_\mathbf{v}\) of a graph \(G\) having adjacency matrix \(\mathbf{A}\) is an \(n\times n\) matrix with columns \(\mathbf{v},\mathbf{A}\mathbf{v},\mathbf{A}^2\mathbf{v},\ldots,\mathbf{A}^{n-1}\mathbf{v}\) whose Gram matrix has constant skew diagonals, each containing walk enumerations in \(G\). We consider the factorization over \(\mathbb{Q}\) of the minimal polynomial \(m(G,x)\) of \(\mathbf{A}\). We prove that the rank of \(\mathbf{W}_\mathbf{v}\), for any walk vector \(\mathbf{v}\), is equal to the sum of the degrees of some, or all, of the polynomial factors of \(m(G,x)\). For some adjacency matrix \(\mathbf{A}\) and a walk vector \(\mathbf{v}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable if \(\mathbf{W}_\mathbf{v}\) has full rank. We show that for graphs having an irreducible characteristic polynomial over \(\mathbb{Q}\), the pair \((\mathbf{A},\mathbf{v})\) is controllable for any walk vector \(\mathbf{v}\). We obtain the number of such graphs on up to ten vertices, revealing that they appear to be commonplace. It is also shown that, for all walk vectors \(\mathbf{v}\), the degree of the minimal polynomial of the largest eigenvalue of \(\mathbf{A}\) is a lower bound for the rank of \(\mathbf{W}_\mathbf{v}\). If the rank of \(\mathbf{W}_\mathbf{v}\) attains this lower bound, then \((\mathbf{A},\mathbf{v})\) is called a recalcitrant pair. We reveal results on recalcitrant pairs and present a graph having the property that \((\mathbf{A},\mathbf{v})\) is neither controllable nor recalcitrant for any walk vector \(\mathbf{v}\).

Graphs Having Most of Their Eigenvalues Shared by a Vertex Deleted Subgraph

“…where b is a 0-1 vector. Usually, b is taken to be j, the vector of all ones [11][12][13][14], but there are exceptions [15][16][17][18]. For every i and j, the entry in the ith row and jth column of W b is equal to the number of walks of length j − 1 that start from vertex i and end at any vertex in S, where S is the subset of V (G) indicated by the entries in b that are equal to 1.…”

“…For every i and j, the entry in the ith row and jth column of W b is equal to the number of walks of length j − 1 that start from vertex i and end at any vertex in S, where S is the subset of V (G) indicated by the entries in b that are equal to 1. It is known (see [15,19]) that, for any indicator vector b and any number of columns k of W b we choose the matrix to have, there is a number r such that the rank of W b is k for all k ≤ r and is r for all k > r. For this reason, W b is either assumed to have r columns [11,15,19], or n columns [12,16,18].…”

“…As in the References [15,16], for each walk matrix W v , we consider the Gram matrix of its columns, which is the matrix…”

“…It is known that if G is a connected graph, then the largest eigenvalue of φ(G, x) is always a root of φ v (x) for any v ∈ V (G) [16]. Hence, Theorem 13 cannot be used for connected graphs whose largest eigenvalue has a minimal polynomial whose degree is at least half of |V (G)|.…”