Minimum rank of graphs that allow loops

Minimum Rank of Graphs with Loops

A loop graph G is a finite undirected graph that allows loops but does not allow multiple edges. The set S(G) of real symmetric matrices associated with a loop graph G of order n is the set of symmetric matrices A = [aij ] ∈ R such that aij 6= 0 if and only if ij ∈ E(G). The minimum (maximum) rank of a loop graph is the minimum (maximum) of the ranks of the matrices in S(G). Loop graphs having ...

Minimum Rank of Graphs That Allow Loops Table of Contents

Ela Minimum Vector Rank and Complement Critical Graphs

Given a graph G, a real orthogonal representation of G is a function from its set of vertices to R such that two vertices are mapped to orthogonal unit vectors if and only if they are not neighbors. The minimum vector rank of a graph is the smallest dimension d for which such a representation exists. This quantity is closely related to the minimum semidefinite rank of G, which has been widely s...

Minimum vector rank and complement critical graphs

Given a graph G, a real orthogonal representation of G is a function from its set of vertices to R such that two vertices are mapped to orthogonal unit vectors if and only if they are not neighbors. The minimum vector rank of a graph is the smallest dimension d for which such a representation exists. This quantity is closely related to the minimum semidefinite rank of G, which has been widely s...

Ela on Bipartite Graphs Which Attain Minimum Rank among Bipartite Graphs with a given Diameter

The rank of a graph is defined to be the rank of its adjacency matrix. In this paper, the bipartite graphs that attain the minimum rank among bipartite graphs with a given diameter are completely characterized.