An upper bound for the minimum rank of a graph

Abstract For a graph G of order n , the minimum rank of G is defined to be the smallest possible rank over all real symmetric n × n matrices A whose ( i , j ) th entry (for i ≠ j ) is nonzero whenever { i , j } is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields.