On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices - Cardoso, Domingos M. (2018)
Let $G$ be a simple undirected graph. Let $0≤ α ≤ 1$. Let $$A_{α}(G)= α D(G) + (1-α) A(G)$$ 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 $m_{G}(α)$ be the multiplicity of $α$ as eigenvalue of $A_{α}(G)$. It is proved that \begin{equation*} m_{G}(α) ≥ 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}(α)= p(G)-q(G)+m_{N}(α) \end{equation*} is determined in which $m_{N}(α)$ is the multiplicity of $α$ as eigenvalue of the matrix $N$. This matrix is obtained from $A_{α}(G)$ 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 eigenvalue of $A_{α}(G)$ 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.