RETRACTED ARTICLE: On ${A_{\alpha }}$-spectrum of a unicyclic graph

Abstract

Let G be a graph of order n with adjacency matrix A(G) and diagonal matrix D(G). For any real $\alpha \in [0,1]$, denote $A_{\alpha }(G):=\alpha D(G)+(1-\alpha ) A(G)$ be ${A_{\alpha }}$-matrix of graph G. The eigenvalues of $A_{\alpha }(G)$ are $\lambda _{1}(A_{\alpha }(G))\ge \lambda _{2}(A_{\alpha }(G))\ge \cdots \ge \lambda _{n}(A_{\alpha }(G))$, the largest eigenvalue $\lambda _{1}(A_{\alpha }(G))$ is called the $A_{\alpha }$-spectral radius of G. The $A_{\alpha }$-separator $S_{A_{\alpha }}(G)$ of graph G is defined as $S_{A_{\alpha }}(G)=\lambda _{1}(A_{\alpha }(G))-\lambda _{2}(A_{\alpha }(G))$. For two disjoint graphs $G_{1}$ and $G_{2}$ (where $V(G_{1})$ and $V(G_{2})$ are disjoint with $v_{1} \in V(G_{1})$, $v_{2} \in V(G_{2})$); the coalescence of $G_{1}$ and $G_{2}$ with respect to $v_{1}$ and $v_{2}$ is formed by identifying $v_{1}$ and $v_{2}$ and is denoted by $G_{1}\cdot G_{2}$. The $A_{\alpha }$-characteristic polynomial of G is defined to be $\Phi (A_{\alpha };x)=det(xI_{n}-A_{\alpha }(G))$, where $I_{n}$ is the identity matrix of size n. A unicyclic graph is a simple connected graph in which the number of edges is equal to the number of vertices. In this paper, firstly, we give the $A_{\alpha }$-characteristic polynomial of the coalescent graph, and $A_{\alpha }$-eigenvalues of the star graph for the application. Secondly, we study the extremal graphs with the maximum and minimum $A_{\alpha }$-spectral radius of the unicyclic graph. Finally, we present the extremal graph with the maximum $A_{\alpha }$-separator of the unicyclic graph and calculate the range of $A_{\alpha }$-separator of the corresponding extremal graph.