Minimum Rank of Graphs with Loops

Abstract

A loop graph $\mf G$ is a finite undirected graph that allows loops but does not allow multiple edges. The set $\sym(\lG)$ of real symmetric matrices associated with a loop graph $\lG$ of order $n$ is the set of symmetric matrices $A=[a_{ij}]\in\Rnn$ such that $a_{ij}\ne 0$ if and only if $ij\in E(\lG)$. The minimum (maximum) rank of a loop graph is the minimum (maximum) of the ranks of the matrices in $\sym(\lG)$. We characterize loop graphs having minimum rank at most two (by forbidden induced subgraphs and graph complements) and loop graphs having minimum rank equal to the order of the graph. A Schur complement reduction technique is used to determine the minimum ranks of cycles with various loop configurations; we also determine the minimum ranks of complete graphs and paths with various configurations of loops. Unlike simple graphs, loop graphs can have maximum rank less than the order of the graph. We present some results on maximum rank and which ranks between minimum and maximum can be realized. Interesting open questions remain.