Minimal Skew energy of oriented bicyclic graphs with a given diameter

TL;DR

This paper identifies the oriented bicyclic graphs with the minimal skew energy for a fixed diameter, considering certain cycle length restrictions, thereby advancing understanding of spectral properties of such graphs.

Contribution

It determines the minimal skew energy oriented bicyclic graphs with a given diameter and specific cycle length constraints, filling a gap in spectral graph theory.

Findings

Identified graphs with minimal skew energy for given diameter and cycle conditions

Extended spectral analysis to bicyclic graphs with orientation constraints

Provided explicit characterization of extremal graphs

Abstract

Let $S(G^{\sigma})$ be the skew-adjacency matrix of the oriented graph $G^{\sigma}$, which is obtained from a simple undirected graph $G$ by assigning an orientation $\sigma$ to each of its edges. The skew energy of an oriented graph $G^{\sigma}$ is defined as the sum of absolute values of all eigenvalues of $S(G^{\sigma})$. For any positive integer $d$ with $3\leq d\leq n-3$, we determine the graph with minimal skew energy among all oriented bicyclic graphs that contain no vertex disjoint odd cycle of lengths $s$ and $l$ with $s+l\equiv 2(mod 4)$ on $n$ vertices with a given diameter $d$.