The sharp upper bounds on the \(A_{\alpha}\)-spectral radius of \(C_4\)-free graphs and Halin graphs
\textit{V. Nikiforov} in [Appl. Anal. Discrete Math. 11, No. 1, 81--107 (2017; Zbl 1499.05384)] studied a combination of the adjacency matrix \(A(G)\) and the signless Laplacian matrix \(Q(G)\) of a graph \(G\) suggested by Dragoš Cvetkovićthe. For this the matrix, \(A_\alpha(G)\) is defined as \[ A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),\quad 0\leqslant \alpha \leqslant1, \] where \(D(G)\) is the diagonal matrix of the degrees of the graph \(G\). Among other problems, the following was proposed: Problem D. Given a graph \(F\), what is the maximum \(\lambda (A_\alpha (G))\) of a graph \(G\) of order \(n\), with no subgraph isomorphic to \(F\). Where \(\lambda (A_\alpha (G))\) is the spectral radius of \(A_\alpha (G)\). The authors solve the above problem when \(F\) is the cicle \(C_4\). Simultaneously, this problem was solved in [\textit{G.-X. Tian} et al., Linear Algebra Appl. 619, 160--175 (2021; Zbl 1462.05245) ]. The authors also give an upper bound for \(\lambda (A_\alpha (G))\) when \(G\) is a Halin graph.
scientific article; zbMATH DE number 7451453
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