Transformations of a graph increasing its Laplacian polynomial and number of spanning trees

Abstract

Let Gmn denote the set of simple graphs with n vertices and m edges, t(G) the number of spanning trees of a graph G, and L(λ, G) the Laplacian polynomial of G. We give some operations Q on graphs such that if G ∈ Gmn then Q(G) ∈ G and L(λ, Q(G) ≤ L(λ, Q(G)) for λ≥n. Because of the relation t(Ks\E(Gn)) = ss-n-2 L(s, Gn) [5], these operations also increase the number of spanning trees of the corresponding complement graphs: t(Ks\E(G)) ≤ t(Ks\E(Q(G)). The approach developed can be used to find some other graph operations with the same property. © 1997 Academic Press Limited.