A characteristic polynomial of the Seidel matrix over $\mathbb{F}_3$

TL;DR

This paper investigates the factorization of Seidel matrix characteristic polynomials over the finite field  and establishes a connection with adjacency matrix polynomials, revealing new algebraic properties of graphs.

Contribution

It characterizes when Seidel matrix polynomials decompose over  and proves their equality with adjacency matrix polynomials over , providing new insights into graph algebraic structures.

Findings

Seidel matrix polynomials decompose into linear factors over  under certain conditions

Characteristic polynomial of Seidel matrix equals that of adjacency matrix over 

New algebraic relations between Seidel and adjacency matrices over 

Abstract

In this paper, we consider the condition when the characteristic polynomials of the Seidel matrix of the graphs are decomposed into products of linear polynomials over $\mathbb{F}_3$. We also show the equality over $\mathbb{F}_3$ between the characteristic polynomial of the Seidel matrix of a graph and the characteristic polynomial of the adjacency matrix of that graph.