Maxima of the \(A_\alpha\)-spectral radius of graphs with given size and minimum degree \(\delta \ge 2\) (Q6625095)

For a graph \(G\), \(A(G)\) denotes its adjacency matrix, and \(D(G)\) denotes the diagonal matrix of its degrees. For any real number \(\alpha \in [0, 1]\), the \(A_{\alpha}\)-matrix of \(G\) is defined as \(A_{\alpha}(G) = \alpha D(G) + (1-\alpha) A(G)\). The largest eigenvalue of \(A_{\alpha}(G)\) is called the \(A_{\alpha}(G)\)-spectral radius of \(G\), denoted by \(\rho_{\alpha}(G)\). In this paper, the author studies the \(A_{\alpha}\)-spectral radius of graphs in terms of a given size \(m\) and minimum degree \(\delta \geq 2\), and characterizes corresponding extremal graphs completely. Furthermore, the author characterizes extremal graphs having maximum \(A_{\alpha}\)-spectral radius among (minimally) 2-edge-connected graphs with given size \(m\).\N\NA friendship graph is one in which every pair of vertices has exactly one common neighbour, denoted by \(F_{\frac{m}{3}}\) for given \(m \equiv 0 \pmod 3\). Let \(\frac{1}{2} \leq \alpha < 1\), and \(G\) be a graph with \(m\) edges and minimum degree \(\delta \geq 2\). The author proves that\N\begin{enumerate}\N\item[(i)] If \(m \geq 24\) and \(m \equiv 0 \pmod 3\), then \(\rho_{\alpha}(G) \leq \rho_{\alpha}(F_{\frac{m}{3}})\), with equality if and only if \(G = F_{\frac{m}{3}}\).\N\item[(ii)] If \(m \geq 37\) and \(m \equiv 1 \pmod 3\), then \(\rho_{\alpha}(G) \leq \rho_{\alpha}(G_{1})\), with equality if and only if \(G = G_{1}\), where \(G_{1} = K_{1} \vee (\frac{m-7}{3} K_{2} \cup K_{1, 3})\).\N\item[(iii)] If \(m \geq 29\) and \(m \equiv 2 \pmod 3\), then \(\rho_{\alpha}(G) \leq \rho_{\alpha}(G_{2})\), with equality if and only if \(G = G_{2}\), where \(G_{2} = K_{1} \vee (\frac{m-5}{3} K_{2} \cup P_{3})\).\N\end{enumerate}\N\NA graph is 2-edge-connected if removing fewer than 2 edges always leaves the remaining graph connected. It is minimally 2-edge-connected if it is 2-edge-connected and deleting any arbitrary chosen edge always leaves a graph that is not 2-edge-connected. For \(m \equiv 1 \pmod 3\), let \(G_{3}\) be the graph obtained from the friendship graph \(F_{\frac{m-1}{3}}\) by subdividing an edge once. For \(m \equiv 2 \pmod 3\), let \(G_{4}\) be the graph obtained from the friendship graph \(F_{\frac{m-2}{3}}\) by subdividing an edge twice.\N\NLet \(\frac{1}{2} \leq \alpha < 1\) and \(G\) be a minimally 2-edge-connected graph with \(m\) edges. The author shows that\N\begin{enumerate}\N\item[(i)] If \(m \geq 24\) and \(m \equiv 0 \pmod 3\), then \(\rho_{\alpha}(G) \leq \rho_{\alpha}(F_{\frac{m}{3}})\), with equality if and only if \(G = F_{\frac{m}{3}}\).\N\item[(ii)] If \(m \geq 37\) and \(m \equiv 1 \pmod 3\), then \(\rho_{\alpha}(G) \leq \rho_{\alpha}(G_{3})\), with equality if and only if \(G = G_{3}\).\N\item[(iii)] If \(m \geq 50\) and \(m \equiv 2 \pmod 3\), then \(\rho_{\alpha}(G) \leq \rho_{\alpha}(G_{4})\), with equality if and only if \(G = G_{4}\).\N\end{enumerate}