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تسجيل مستخدم جديدOn the $A_alpha$ spectral radius and $A_alpha$ energy of non-strongly connected digraphs
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Let $A_alpha(G)$ be the $A_alpha$-matrix of a digraph $G$ and $lambda_{alpha 1}, lambda_{alpha 2}, ldots, lambda_{alpha n}$ be the eigenvalues of $A_alpha(G)$. Let $rho_alpha(G)$ be the $A_alpha$ spectral radius of $G$ and $E_alpha(G)=sum_{i=1}^n lambda_{alpha i}^2$ be the $A_alpha$ energy of $G$ by using second spectral moment. Let $mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal $A_alpha$ spectral radius and the maximal (minimal) $A_alpha$ energy in $mathcal{G}_n^m$.
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Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the degrees of undi
rected graphs. Liu et al. cite{LWCL} extended the definition to digraphs. For any real $alphain[0,1]$, the matrix $A_alpha(G)$ of a digraph $G$ is defined as $$A_alpha(G)=alpha D(G)+(1-alpha)A(G).$$ The largest modulus of the eigenvalues of $A_alpha(G)$ is called the $A_alpha$ spectral radius of $G$, denoted by $lambda_alpha(G)$. This paper proves some extremal results about the spectral radius $lambda_alpha(G)$ that generalize previous results about $lambda_0(G)$ and $lambda_{frac{1}{2}}(G)$. In particular, we characterize the extremal digraph with the maximum (or minimum) $A_alpha$ spectral radius among all $widetilde{infty}$-digraphs and $widetilde{theta}$-digraphs on $n$ vertices. Furthermore, we determine the digraphs with the second and the third minimum $A_alpha$ spectral radius among all strongly connected bicyclic digraphs. For $0leqalphaleqfrac{1}{2}$, we also determine the digraphs with the second, the third and the fourth minimum $A_alpha$ spectral radius among all strongly connected digraphs on $n$ vertices. Finally, we characterize the digraph with the minimum $A_alpha$ spectral radius among all strongly connected bipartite digraphs which contain a complete bipartite subdigraph.
Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $alphainleft[ 0,1right] $, write $A_{alpha}left( Gright) $ for the matrix [ A_{alpha}left( Gright) =alpha Dleft( Gright) +(1-a
lpha)Aleft( Gright) . ] Let $alpha_{0}left( Gright) $ be the smallest $alpha$ for which $A_{alpha}(G)$ is positive semidefinite. It is known that $alpha_{0}left( Gright) leq1/2$. The main results of this paper are: (1) if $G$ is $d$-regular then [ alpha_{0}=frac{-lambda_{min}(A(G))}{d-lambda_{min}(A(G))}, ] where $lambda_{min}(A(G))$ is the smallest eigenvalue of $A(G)$; (2) $G$ contains a bipartite component if and only if $alpha_{0}left( Gright) =1/2$; (3) if $G$ is $r$-colorable, then $alpha_{0}left( Gright) geq1/r$.
Let $G$ be a graph on $n$ vertices, its adjacency matrix and degree diagonal matrix are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov cite{0007} introduced the matrix $A_{alpha}(G)=alpha D(G)+(1-alpha)A(G)$ for $alphain [0, 1].$ The
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