Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are cospectral if and only if the isospectral reduction over these vertices has a nontrivial automorphism. It is well known that if two vertices of a graph are symmetric, i.e. if there exists a graph automorphism permuting these two vertices, then they are cospectral. This paper extends this result showing that any two cospectral vertices are symmetric in some reduced version of the graph. We also prove that two vertices are strongly cospectral if and only if they are cospectral and the isospectral reduction over these two vertices has simple eigenvalues. We further describe how these results can be used to construct new families of graphs with cospectral vertices. *
A k-ranking of a graph G is a labeling of the vertices of G with values from {1, . . . , k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for which a k-ranking of G exists. The graph G is k-critical if it has tree-depth k and every proper minor of G has smaller tree-depth.We establish partial results in support of two conjectures about the order and maximum degree of k-critical graphs. As part of these results, we define a graph G to be 1-unique if for every vertex v in G, there exists an optimal ranking of G in which v is the unique vertex with label 1. We show that several classes of k-critical graphs are 1-unique, and we conjecture that the property holds for all k-critical graphs. Generalizing a previously known construction for trees, we exhibit an inductive construction that uses 1-unique k-critical graphs to generate large classes of critical graphs having a given tree-depth.
A sign pattern matrix is a matrix whose entries are from the set {+, −, 0}. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. It is shown in this paper that for any m×n sign pattern A with minimum rank n − 2, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer n ≥ 9, there exists a nonnegative integer m such that there exists an n × m sign pattern matrix with minimum rank n − 3 for which rational realization is not possible. A characterization of m × n sign patterns A with minimum rank n − 1 is given (which solves an open problem in Brualdi et al.[6]), along with a more general description of sign patterns with minimum rank r, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of k-dimensional subspaces of R n are obtained. In particular, it is shown that the maximum number of sign vectors of 2-dimensional subspaces of R n is 4n + 1. Several related open problems are stated along the way.
A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning forest with 2 components such that u and v are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices u and v in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating u and v divided by the number of spanning trees in the graph. There are also well-known matrix theoretic methods for calculating resistance distance, but the way in which the structure of the underlying graph determines resistance distance via these methods is not well understood.For any connected graph G with a 2-separator separating vertices u and v, we show that the number of spanning trees and spanning 2-forests separating u and v can be expressed in terms of these same quantities for the smaller separated graphs, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if u and v are in the same smaller graph. In this paper we demonstrate that this method of calculating resistance distance is more suitable for certain structured families of graphs than the more standard methods. We apply our results to count the number of spanning 2-forests and calculate the resistance distance in a family of Sierpinski triangles and in the family of linear 2-trees with a single bend.
Abstract. Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n × n matrices whose nonzero off-diagonal entries correspond to the edges of G. Given 2n − 1 real numbers λ 1 ≥ µ 1 ≥ λ 2 ≥ µ 2 ≥ · · · ≥ λ n−1 ≥ µ n−1 ≥ λn, and a vertex v of G, the question is addressed of whether or not there exists A ∈ S(G) with eigenvalues λ 1 , . . . , λn such that A(v) has eigenvalues µ 1 , . . . , µ n−1 , where A(v) denotes the matrix with vth row and column deleted. A complete solution can be given for the path on n vertices with v a pendant vertex and also for the star on n vertices with v the dominating vertex. The main result is a complete solution to this "λ, µ" problem for all connected graphs on 4 vertices.
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