Ordering of graphs with fixed size and diameter by <i>A</i> _α -spectral radii (Q6665958)

The \(A_{\alpha}\)-matrix of a graph \( G\) is defined as the convex linear combination of the adjacency matrix \(A(G)\) and the diagonal matrix of degrees \(D(G)\), i.e. \(A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)\) with \(\alpha \in [0, 1]\). The maximum modulus among all \(A_{\alpha}\)-eigenvalues is called the \(A_{\alpha}\)-spectral radius. In this paper, the authors order the connected graphs with size \(m\) and diameter (at least) \(d\) from the second to the \((\lfloor d/2 \rfloor + 1)\)th regarding the \(A_{\alpha}\)-spectral radius for \( \alpha \in [ \frac{1}{2},1)\). As by-products, the first \(\lfloor d/2\rfloor\) largest trees of order \(n\) and diameter (at least) \(d\) in terms of their \(A_{\alpha}\)-spectral radii are identified, and the unique graph with at least one cycle having the largest \(A_{\alpha}\)-spectral radius among graphs of size \(m\) and diameter (at least) \(d\) is characterized. Consequently, the corresponding results for the signless Laplacian matrix can be deduced as well. An open problem concludes the paper.