21 research outputs found

    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

    On the α\alpha-spectral radius of hypergraphs

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    For real α∈[0,1)\alpha\in [0,1) and a hypergraph GG, the α\alpha-spectral radius of GG is the largest eigenvalue of the matrix Aα(G)=αD(G)+(1−α)A(G)A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G), where A(G)A(G) is the adjacency matrix of GG, which is a symmetric matrix with zero diagonal such that for distinct vertices u,vu,v of GG, the (u,v)(u,v)-entry of A(G)A(G) is exactly the number of edges containing both uu and vv, and D(G)D(G) is the diagonal matrix of row sums of A(G)A(G). We study the α\alpha-spectral radius of a hypergraph that is uniform or not necessarily uniform. We propose some local grafting operations that increase or decrease the α\alpha-spectral radius of a hypergraph. We determine the unique hypergraphs with maximum α\alpha-spectral radius among kk-uniform hypertrees, among kk-uniform unicyclic hypergraphs, and among kk-uniform hypergraphs with fixed number of pendant edges. We also determine the unique hypertrees with maximum α\alpha-spectral radius among hypertrees with given number of vertices and edges, the unique hypertrees with the first three largest (two smallest, respectively) α\alpha-spectral radii among hypertrees with given number of vertices, the unique hypertrees with minimum α\alpha-spectral radius among the hypertrees that are not 22-uniform, the unique hypergraphs with the first two largest (smallest, respectively) α\alpha-spectral radii among unicyclic hypergraphs with given number of vertices, and the unique hypergraphs with maximum α\alpha-spectral radius among hypergraphs with fixed number of pendant edges

    Spectral properties of digraphs with a fixed dichromatic number

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    The A<sub>α</sub> spectral moments of digraphs with a given dichromatic number

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    The Aα-matrix of a digraph G is defined as Aα(G)=αD+(G)+(1−α)A(G), where α∈[0,1), D+(G) is the diagonal outdegree matrix and A(G) is the adjacency matrix. The k-th Aα spectral moment of G is defined as ∑i=1 nλαi k, where λαi are the eigenvalues of the Aα-matrix of G, and k is a nonnegative integer. In this paper, we obtain the digraphs which attain the minimal and maximal second Aα spectral moment (also known as the Aα energy) within classes of digraphs with a given dichromatic number. We also determine sharp bounds for the third Aα spectral moment within the special subclass which we define as join digraphs. These results are related to earlier results about the second and third Laplacian spectral moments of digraphs.</p
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