TWO REMARKS ON THE ADJOINT POLYNOMIAL

AbstractOne can define the adjoint polynomial of the graph G as follows. Let ak(G) denote the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. Then the adjoint polynomial of G is h(G,x)=∑k=1n(−1)n−kak(G)xk, where n denotes the number of vertices of the graph G. In this paper we show that the largest real root γ(G) of h(G,x) has the largest absolute value among the roots. We also prove that γ(G)≤4(Δ−1), where Δ denotes the largest degree of the graph G. This bound is sharp

One more remark on the adjoint polynomial

The adjoint polynomial of G is h(G,x)=∑k=1n(−1)n−kak(G)xk,where ak(G) denotes the number of ways one can cover all vertices of the graph G by exactly k disjoint cliques of G. In this paper we show the adjoint polynomial of a graph G is a simple transformation of the independence polynomial of another graph Ĝ. This enables us to use the rich theory of independence polynomials to study the adjoint polynomials. In particular we give new proofs of several theorems of R. Liu and P. Csikvári. © 2017 Elsevier Lt