کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4600227 | 1336841 | 2012 | 13 صفحه PDF | دانلود رایگان |
In this paper we introduce a new parameter for a graph called the 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 formU(α,β,γ,δ)=αA+βI+γJ+δDU(α,β,γ,δ)=αA+βI+γJ+δDwhere 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≠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.
Journal: Linear Algebra and its Applications - Volume 437, Issue 8, 15 October 2012, Pages 2064–2076