An improved lower bound for the Seidel energy of tree graphs

TL;DR

This paper improves the lower bound of Seidel energy specifically for tree graphs, refining previous general bounds and providing more precise estimates for this class of graphs.

Contribution

The paper introduces a tighter lower bound for the Seidel energy of tree graphs, advancing the understanding of spectral properties of these structures.

Findings

Established a new lower bound for Seidel energy of trees

Compared the new bound with previous general bounds

Validated the bound through theoretical analysis

Abstract

Let $G$ be a graph with the vertex set $ \lbrace v_1,\ldots,v_n \rbrace$. The Seidel matrix of $G$ is an $n\times n$ matrix whose diagonal entries are zero, $ij$-th entry is $-1$ if $ v_{i} $ and $ v_{j} $ are adjacent and otherwise is $ 1 $. The Seidel energy of $G$, denoted by $ \se{G} $, is defined to be the sum of absolute values of all eigenvalues of the Seidel matrix of $G$. In \cite{aekn}, the authors proved that the Seidel energy of any graph of order $n$ is at least $2n-2$. In this study, we improve the aforementioned lower bound for tree graphs.