Related papers: Sharp spectral bounds for the edge-connectivity of…

Connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol proposed the $g$-good-neighbor connectivity of $G$. In this paper, we show that $1\leq…

It is shown that for a given infinite graph $G$ on countably many vertices, and a compact, infinite set of real numbers $\Lambda$ there is a real symmetric matrix $A$ whose graph is $G$ and its spectrum is $\Lambda$. Moreover, the set of…

Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently,…

The $(\kappa,\ell)$-edge-inducibility problem asks for the maximum number of $\kappa$-subsets inducing exactly $\ell$ edges that a graph of given order $n$ can have. Using flag algebras and stability approach, we resolve this problem for…

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real $\alpha\in[0,1]$. The largest eigenvalue of $A_\alpha(G)$ is…

Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $\lambda_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for…

Signed graphs have their edges labeled either as positive or negative. $\rho(M)$ denote the $M$-spectral radius of $\Sigma$, where $M=M(\Sigma)$ is a real symmetric graph matrix of $\Sigma$. Obviously,…

For a graph $G$ of order $n$, let $$ \lambda_1(G)\ge \cdots \ge \lambda_n(G) $$ be the eigenvalues of its adjacency matrix. We prove that every graph $G$ on $n\ge 3$ vertices satisfies $$ \lambda_3(G)\le \frac{n}{3}-1, $$ thereby solving a…

In this paper a tight lower bound for algebraic connectivity of graphs (second smallest eigenvalue of the Laplacian matrix of the graph) based on connection-graph-stability method is introduced. The connection-graph-stability score for each…

For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is…

The difference between the two largest eigenvalues of the adjacency matrix of a graph $G$ is called the spectral gap of $G.$ If $G$ is a regular graph, then its spectral gap is equal to algebraic connectivity. Abdi, Ghorbani and Imrich, in…

Jord\'an and Tanigawa recently introduced the $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G$. This is a quantitative measure of the $d$-dimensional rigidity of $G$ which generalizes the well-studied notion of spectral…

For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $\lambda(G)$ denotes…

For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the diagonal matrix of vertex degrees of $G$,…

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…

For any $\epsilon > 0$, we show that if $G$ is a regular graph on $n \gg_\epsilon 1$ vertices that is $\epsilon$-far (differs by at least $\epsilon n^2$ edges) from any Tur\'{a}n graph, then its second eigenvalue $\lambda_2$ satisfies…

From Alon and Boppana, and Serre, we know that for any given integer $k\geq 3$ and real number $\lambda<2\sqrt{k-1}$, there are finitely many $k$-regular graphs whose second largest eigenvalue is at most $\lambda$. In this paper, we…

A graph is minimally $k$-connected ($k$-edge-connected) if it is $k$-connected ($k$-edge-connected) and deleting arbitrary chosen edge always leaves a graph which is not $k$-connected ($k$-edge-connected). A classic result of minimally…

We prove that for each $d\geq 3$ and $k\geq 2$, the set of limit points of the first $k$ eigenvalues of sequences of $d$-regular graphs is \[ \{(\mu_1,\dots,\mu_k): d=\mu_1\geq \dots\geq \mu_{k}\geq2\sqrt{d-1}\}. \] The result for $k=2$ was…

We prove that for each $d \geq 3$ the set of all limit points of the second largest eigenvalue of growing sequences of $d$-regular graphs is $[2\sqrt{d-1},d]$. A similar argument shows that the set of all limit points of the smallest…