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    On the maximum AαA_{\alpha}-spectral radius of unicyclic and bicyclic graphs with fixed girth or fixed number of pendant vertices

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    For a connected graph GG, let A(G)A(G) be the adjacency matrix of GG and D(G)D(G) be the diagonal matrix of the degrees of the vertices in GG. The AαA_{\alpha}-matrix of GG is defined as \begin{align*} A_\alpha (G) = \alpha D(G) + (1-\alpha) A(G) \quad \text{for any α∈[0,1]\alpha \in [0,1]}. \end{align*} The largest eigenvalue of Aα(G)A_{\alpha}(G) is called the AαA_{\alpha}-spectral radius of GG. In this article, we characterize the graphs with maximum AαA_{\alpha}-spectral radius among the class of unicyclic and bicyclic graphs of order nn with fixed girth gg. Also, we identify the unique graphs with maximum AαA_{\alpha}-spectral radius among the class of unicyclic and bicyclic graphs of order nn with kk pendant vertices.Comment: 16 page
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