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Let $mathcal{G}$ be an undirected graph with adjacency matrix $A$ and spectral radius $rho$. Let $w_k, phi_k$ and $phi_k^{(i)}$ be, respectively, the number walks of length $k$, closed walks of length $k$ and closed walks starting and ending at vertex $i$ after $k$ steps. In this paper, we propose a measure-theoretic framework which allows us to relate walks in a graph with its spectral properties. In particular, we show that $w_k, phi_k$ and $phi_k^{(i)}$ can be interpreted as the moments of three different measures, all of them supported on the spectrum of $A$. Building on this interpretation, we leverage results from the classical moment problem to formulate a hierarchy of new lower and upper bounds on $rho$, as well as provide alternative proofs to several well-known bounds in the literature.
In this paper, we present two sharp upper bounds for the spectral radius of (bipartite) graphs with forbidden a star forest and characterize all extremal graphs. Moreover, the minimum least eigenvalue of the adjacency matrix of graph with forbidden a
Let $G$ be a simple graph with vertex set $V(G) = {v_1 ,v_2 ,cdots ,v_n}$. The Harary matrix $RD(G)$ of $G$, which is initially called the reciprocal distance matrix, is an $n times n$ matrix whose $(i,j)$-entry is equal to $frac{1}{d_{ij}}$ if $i ot
Let $A(G)$ be the adjacency matrix of a graph $G$ with $lambda_{1}(G)$, $lambda_{2}(G)$, ..., $lambda_{n}(G)$ being its eigenvalues in non-increasing order. Call the number $S_k(G):=sum_{i=1}^{n}lambda_{i}^k(G) (k=0,1,...,n-1)$ the $k$th spectral mom
We obtain new bounds for the Laplacian spectral radius of a signed graph. Most of these new bounds have a dependence on edge sign, unlike previously known bounds, which only depend on the underlying structure of the graph. We then use some of these b
Write $rholeft( Gright) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}veeoverline{K}_{n-r}.$ Let $n>rgeq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the