Transformations of a Graph Increasing its Laplacian Polynomial and Number of Spanning Trees

LetG rmndenote the set of simple graphs withnvertices andmedges,t(G)the number of spanning trees of a graphG,andL(λ, G)the Laplacian polynomial ofG.We give some operationsQon graphs such that ifG∈G rmnthenQ(G)∈G rmnandL(λ, G)≤L(λ, Q(G))forλ≤n.Because of the relationt(Ks\E(Gn)) =srs-n-2L(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.