On the multiplicity of α as an eigenvalue of the Aα matrix of a graph in terms of the number of pendant vertices

Let G=(V(G),E(G)) be a simple undirected graph with vertex set V(G) and edge set E(G). The cyclomatic number of a connected graph G is defined as θ(G)=|E(G)|−|V(G)|+1. The Aα matrix of a graph G is defined by Nikiforov as Aα(G)=αD(G)+(1−α)A(G), where α∈[0,1], A(G) and D(G) respectively denotes the adjacency matrix and the diagonal matrix of the vertex degrees of G. Let mG(λ) be the multiplicity of λ as an eigenvalue of Aα(G). A cluster of G is an independent set of one or more vertices of G, each of which has the same set of neighbors. The closure of a cluster is the set of vertices of the cluster together with all the neighbors of the cluster. The degree of a cluster is the cardinality of its shared set of neighbors. A k-cluster is a cluster of degree k. The number of vertices in a k-cluster is its order. A collection of two or more k-clusters is independent if the k-clusters are pairwise disjoint and is isolated if the closures of the k-clusters are pairwise disjoint. Cardoso et al. [2] obtained a lower bound for mG(α) in terms of the number of pendant vertices. They proved that, for a simple connected graph G with p(G)>0 pendant vertices attached at q(G) quasi-pendant vertices, mG(α)≥p(G)−q(G), with equality if each internal vertex is a quasi-pendant vertex. In this article, for a graph G of order n and with t independent k-clusters of orders r1,…,rt, we prove that mG(kα)≥∑i=1tri−t, and if the k-clusters are isolated, then mG(kα)≤n−2t, which extends Cardoso's result to a more general case. Also, we give an upper bound for mG(α) in terms of p(G). It is proved that mG(α)≤p(G)+2θ(G) with θ(G) the cyclomatic number of G, the graphs G whose mG(α) attain the upper bound are characterized explicitly.