Related papers: Multiplicity Bounds for Arbitrary Eigenvalues of C…

We consider the normalized adjacency matrix of a random $d$-regular graph on $N$ vertices with any fixed degree $d\geq 3$ and denote its eigenvalues as $\lambda_1=d/\sqrt{d-1}\geq \lambda_2\geq\lambda_3\cdots\geq \lambda_N$. We establish…

Let $G$ be a connected simple graph of order $n$. Let $\rho_1(G)\geq \rho_2(G)\geq \cdots \geq \rho_{n-1}(G)> \rho_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(\rho_i)$ the multiplicity…

Graph theory on surfaces extends classical graph structures to topological surfaces, providing a theoretical foundation for characterizing the embedding properties of complex networks in constrained spaces. The study of bounding the…

The number of embeddings of minimally rigid graphs in $\mathbb{R}^D$ is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap…

Let $G$ be a connected graph on $n$ vertices with girth $g$. Let $m_GI$ denote the number of Laplacian eigenvalues of graph $G$ in an interval $I$. In this paper, we show that if $G$ is not a cycle, then $m_G(n-g+3,n]\leq n-g$. Moreover, we…

The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity…

Let $\mathcal A$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by $\lambda_1\geq \lambda_2\cdots \geq \lambda_{N}$. For $N^{2/3}\ll d\leq N/2$, we prove optimal rigidity estimates of…

For a connected graph $G$ with order $n$, let $e(G)$ be the number of its distinct eigenvalues and $d$ be the diameter. We denote by $m_G(\mu)$ the eigenvalue multiplicity of $\mu$ in $G$. It is well known that $e(G)\geq d+1$, which shows…

Let $\Gamma=(K_{n},H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. The index of $\Gamma$ is the largest eigenvalue of its adjacency matrix. In this paper we study the index of $\Gamma$ when $H$ is a unicyclic…

A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi = (G, \varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation.…

Let $\lambda_2(G)$ and $\kappa'(G)$ be the second largest eigenvalue and the edge-connectivity of a graph $G$, respectively. Let $d$ be a positive integer at least 3. For $t=1$ or 2, Cioaba proved sharp upper bounds for $\lambda_2(G)$ in a…

Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a…

In this paper we explore some results concerning the spread of the line and the total graph of a given graph. In particular, it is proved that for an $(n,m)$ connected graph $G$ with $m > n \geq 4$ the spread of $G$ is less than or equal to…

A mixed graph $\widetilde{G}$ is obtained from a simple undirected graph $G$, the underlying graph of $\widetilde{G}$, by orienting some edges of $G$. Let $c(G)=|E(G)|-|V(G)|+\omega(G)$ be the cyclomatic number of $G$ with $\omega(G)$ the…

Let $G$ be a connected $d$-regular graph of order $n$, where $d\geq3$. Let $\lambda_{2}(G)$ be the second largest eigenvalue of $G$. For even $n$, we show that $G$ contains $\left\lfloor\frac{2}{3}(d-\lambda_{2}(G))\right\rfloor$…

Let $m(G,\lambda)$ be the multiplicity of an eigenvalue $\lambda$ of a connected graph $G$. Wang et al. [Linear Algebra Appl. 584(2020), 257-266] proved that for any connected graph $G\neq C_n$, $m(G, \lambda) \leq 2c(G) + p(G) -1$, where…

A signed tree-coloring of a signed graph $(G,\sigma)$ is a vertex coloring $c$ so that $G^{c}(i,\pm)$ is a forest for every $i\in c(u)$ and $u\in V(G)$, where $G^{c}(i,\pm)$ is the subgraph of $(G,\sigma)$ whose vertex set is the set of…

Let $\lambda(G)$ be the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. Let $\lambda_{\rm e}(G)$ be the smallest number of edges that can be removed from…

A signed graph $(G,\sigma)$ is a graph $G$ with a signature $\sigma$ labeling each edge with a positive or negative sign. Two signatures of $G$ are switching equivalent if one is obtained from the other by changing the signs of all edges in…

The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. Recently a new characterization of…