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

Abstract

Let $G$ be a simple undirected graph. Let $0\leq \alpha \leq 1$. Let $$A_{\alpha}(G)= \alpha D(G) + (1-\alpha) A(G)$$ 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 $m_{G}(\alpha)$ be the multiplicity of $\alpha$ as eigenvalue of $A_{\alpha}(G)$. It is proved that \begin{equation*} m_{G}(\alpha) \geq p(G) - q(G) \end{equation*} 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 equality \begin{equation*} m_{G}(\alpha)= p(G)-q(G)+m_{N}(\alpha) \end{equation*} is determined in which $m_{N}(\alpha)$ is the multiplicity of $\alpha$ as eigenvalue of the matrix $N$. This matrix is obtained from $A_{\alpha}(G)$ taking the entries corresponding to the internal vertices which are non quasi-pendant vertices. These results are applied to search for the multiplicity of $\alpha$ as eigenvalue of $A_{\alpha}(G)$ 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.