RETRACTED ARTICLE: On $${A_{\alpha }}$$ A α -spectrum of a unicyclic graph

Let G be a graph of order n with adjacency matrix A(G) and diagonal matrix D(G). For any real $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] , denote $$A_{\alpha }(G):=\alpha D(G)+(1-\alpha ) A(G)$$ A α ( G ) : = α D ( G ) + ( 1 - α ) A ( G ) be $${A_{\alpha }}$$ A α -matrix of graph G. The eigenvalues of $$A_{\alpha }(G)$$ A α ( G ) are $$\lambda _{1}(A_{\alpha }(G))\ge \lambda _{2}(A_{\alpha }(G))\ge \cdots \ge \lambda _{n}(A_{\alpha }(G))$$ λ 1 ( A α ( G ) ) ≥ λ 2 ( A α ( G ) ) ≥ ⋯ ≥ λ n ( A α ( G ) ) , the largest eigenvalue $$\lambda _{1}(A_{\alpha }(G))$$ λ 1 ( A α ( G ) ) is called the $$A_{\alpha }$$ A α -spectral radius of G. The $$A_{\alpha }$$ A α -separator $$S_{A_{\alpha }}(G)$$ S A α ( G ) of graph G is defined as $$S_{A_{\alpha }}(G)=\lambda _{1}(A_{\alpha }(G))-\lambda _{2}(A_{\alpha }(G))$$ S A α ( G ) = λ 1 ( A α ( G ) ) - λ 2 ( A α ( G ) ) . For two disjoint graphs $$G_{1}$$ G 1 and $$G_{2}$$ G 2 (where $$V(G_{1})$$ V ( G 1 ) and $$V(G_{2})$$ V ( G 2 ) are disjoint with $$v_{1} \in V(G_{1})$$ v 1 ∈ V ( G 1 ) , $$v_{2} \in V(G_{2})$$ v 2 ∈ V ( G 2 ) ); the coalescence of $$G_{1}$$ G 1 and $$G_{2}$$ G 2 with respect to $$v_{1}$$ v 1 and $$v_{2}$$ v 2 is formed by identifying $$v_{1}$$ v 1 and $$v_{2}$$ v 2 and is denoted by $$G_{1}\cdot G_{2}$$ G 1 · G 2 . The $$A_{\alpha }$$ A α -characteristic polynomial of G is defined to be $$\Phi (A_{\alpha };x)=det(xI_{n}-A_{\alpha }(G))$$ Φ ( A α ; x ) = d e t ( x I n - A α ( G ) ) , where $$I_{n}$$ 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 $$A_{\alpha }$$ A α -characteristic polynomial of the coalescent graph, and $$A_{\alpha }$$ A α -eigenvalues of the star graph for the application. Secondly, we study the extremal graphs with the maximum and minimum $$A_{\alpha }$$ A α -spectral radius of the unicyclic graph. Finally, we present the extremal graph with the maximum $$A_{\alpha }$$ A α -separator of the unicyclic graph and calculate the range of $$A_{\alpha }$$ A α -separator of the corresponding extremal graph.