On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices

Let G be a simple undirected graph. Let 0≤α≤1. LetAα(G)=αD(G)+(1−α)A(G) where D(G) and A(G) are the diagonal matrix of the vertex degrees of G and the adjacency matrix of G, respectively. Let p(G)>0 and q(G) be the number of pendant vertices and quasi-pendant vertices of G, respectively. Let mG(α) be the multiplicity of α as an eigenvalue of Aα(G). It is proved thatmG(α)≥p(G)−q(G) with equality if each internal vertex is a quasi-pendant vertex. If there is at least one internal vertex which is not a quasi-pendant vertex, the equalitymG(α)=p(G)−q(G)+mN(α) is determined in which mN(α) is the multiplicity of α as an eigenvalue of the matrix N. This matrix is obtained from Aα(G) taking the entries corresponding to the internal vertices which are non quasi-pendant vertices. These results are applied to search for the multiplicity of α as an eigenvalue of Aα(G) when G is a path, a caterpillar, a circular caterpillar, a generalized Bethe tree or a Bethe tree. For the Bethe tree case, a simple formula for the nullity is given.