Theses and Dissertations

Issuing Body

Mississippi State University

Advisor

Sivaraman, Vaidyanathan

Committee Member

Fabel, Paul

Committee Member

Qian, Chuanxi

Committee Member

Yarahmadian, Shantia

Date of Degree

8-8-2023

Document Type

Dissertation - Campus Access Only

Major

Mathematical Science

Degree Name

Doctor of Philosophy (Ph.D)

College

College of Arts and Sciences

Department

Department of Mathematics and Statistics

Abstract

In this dissertation, we investigate various aspects of signed graphs, with a particular focus on signings and sign-symmetric signed graphs. We begin by examining the complete graph on six vertices with one edge deleted ($K_6$\textbackslash e) and explore the different ways of signing this graph up to switching isomorphism. We determine the frustration index (number) of these signings and investigate the existence of sign-symmetric signed graphs. We then extend our study to the $K_6$\textbackslash 2e graph and the McGee graph with exactly two negative edges. We investigate the distinct ways of signing these graphs up to switching isomorphism and demonstrate the absence of sign-symmetric signed graphs in some cases. We then introduce and study the signed graph class $\mathcal{S}$, which includes all sign-symmetric signed graphs, we prove several theorems and lemmas as well as discuss the class of tangled sign-symmetric signed graphs. Also, we study the graph class $\mathcal{G}$, consisting of graphs with at least one sign-symmetric signed graph, prove additional theorems and lemmas, and determine certain families within $\mathcal{G}$. Our results have practical applications in various fields such as social psychology and computer science.

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