The minimum rank of universal adjacency matrices

Abstract

In this paper we introduce a new parameter for a graph called the {\it minimum universal rank}. This parameter is similar to the minimum rank of a graph. For a graph $G$ the minimum universal rank of $G$ is the minimum rank over all matrices of the form \[ U(α, β, γ, δ) = αA + βI + γJ + δD \] where $A$ is the adjacency matrix of $G$, $J$ is the all ones matrix and $D$ is the matrix with the degrees of the vertices in the main diagonal, and $α\neq 0, β, γ, δ$ are scalars. Bounds for general graphs based on known graph parameters are given, as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of $A$. The exact value of the minimum universal rank of some families of graphs are determined, including complete graphs, complete bipartite graph, paths and cycles. Bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established. It is shown that the minimum universal rank is not monotone on induced subgraphs, but bounds based on certain induced subgraphs, including bounds on the union of two graphs, are given. Finally we characterize all graphs with minimum universal rank equal to 0 and to 1.