On Szeged Polynomial of Graphs with Even Number of Vertices

Abstract

The Szeged polynomial of a connected graph \(G\) is defined as \(S_z(G,x) = \sum_{e \in E(G)} x^{n_{u(e) n_v(e)}} \), where \(n_u(e)\) is the number of vertices of \(G\) lying closer to \(u\) than to \(v\), and \(n_v(e)\) is the number of vertices of \(G\) lying closer to \(v\) than to \(u\). Ashrafi et al. (On Szeged polynomial of a graph, Bull. Iran. Math. Soc. \(33 (2007) 37-46)\) proved that if \(|V(G)|\) is even, then \(\deg(S_z(G,x)) \leq \frac{1}{4}{|V(G)^{2}} |\). In this paper, we investigate the structure of graphs with an even number of vertices for which equality holds, and also examine equality for the sum of graphs.