On Szeged polynomial of graphs with even number of Vertices

Abstract

The Szeged polynomial of a connected graph G, is defined as Sz(G,x) = σ eεE(G)xnu (e)nv(e) where nu(e) is the number of vertices of G lying closer to u than to ν, nu(e) is the number of vertices of G lying closer to ν than to u and the summation goes over all edges e = uν ε E(G) of G. Ashrafi et. al. (On Szeged polynomial of a graph, Bui. Iran. Math. Soc. 33 (2007) 37-46.) proved that if the number of the vertices of G is even, then deg(Sz(G, x)) < 1/4 [V(G)2] where V(G) is the set of vertices of G. In this paper we study the structure of graphs, with even number of vertices, for which the equality holds. Also we examine equality for the sum of graphs. Copyright © 2014, Charles Babbage Research Centre.