A note on the minimum skew rank of a graph

TL;DR

This paper explores properties of the minimum skew rank of graphs over fields, providing new characterizations, exact values for specific graph classes, and extending known results relating skew rank to graph matchings.

Contribution

It introduces new properties and characterizations of the minimum skew rank, including for graphs with cut vertices, k-paths, and graphs without even cycles, extending existing theoretical results.

Findings

Characterization of graphs with cut vertices and skew rank 4 over infinite fields

Determination of minimum skew rank for k-paths

Equality of minimum skew rank, matching number, and maximum skew rank for certain graphs

Abstract

The minimum skew rank $mr^{-}(\mathbb{F},G)$ of a graph $G$ over a field $\mathbb{F}$ is the smallest possible rank among all skew symmetric matrices over $\mathbb{F}$, whose ($i$,$j$)-entry (for $i\neq j$) is nonzero whenever $ij$ is an edge in $G$ and is zero otherwise. We give some new properties of the minimum skew rank of a graph, including a characterization of the graphs $G$ with cut vertices over the infinite field $\mathbb{F}$ such that $mr^{-}(\mathbb{F},G)=4$, determination of the minimum skew rank of $k$-paths over a field $\mathbb{F}$, and an extending of an existing result to show that $mr^{-}(\mathbb{F},G)=2match(G)=MR^{-}(\mathbb{F},G)$ for a connected graph $G$ with no even cycles and a field $\mathbb{F}$, where $match(G)$ is the matching number of $G$, and $MR^{-}(\mathbb{F},G)$ is the largest possible rank among all skew symmetric matrices over $\mathbb{F}$.