A graph in which every edge is labeled positive or negative is called a signed graph. We determine the number of ways to sign the edges of the McGee graph with exactly two negative edges up to switching isomorphism. We characterize signed graphs that are both sign-symmetric and have a frustration index of 1. We prove some results about which signed graphs on complete multipartite graphs have frustration indices 2 and 3. In the final part, we derive the relationship between the frustration index and the number of parts in a sign-symmetric signed graph on complete multipartite graphs.

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.