Related papers: A best bound for $\lambda_2(G)$ to guarantee $\kap…

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…

An L(2,1)-labelling of a graph $G=(V, E)$ is $\lambda_{2,1}(G)$ a function $f$ from the vertex set V (G) to the set of non-negative integers such that adjacent vertices get numbers at least two apart, and vertices at distance two get…

One of the best-known results in spectral graph theory is the inequality of Hoffman \[ \chi\left( G\right) \geq1-\frac{\lambda\left( G\right) }{\lambda_{\min }\left( G\right) }, \] where $\chi\left( G\right) $ is the chromatic number of a…

In 2002, Nikiforov proved that for an $n$-vertex graph $G$ with clique number $\omega$ and edge number $m$, the spectral radius $\lambda(G)$ satisfies $\lambda (G) \leq \sqrt{2(1 - 1/\omega) m}$, which confirmed a conjecture implicitly…

The study of eigenvalue multiplicities plays a central role in the spectral theory of signed graphs, extending several classical results from the unsigned setting. While most existing work focuses on the nullity of a signed graph (the…

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes…

Let $\Delta$ and $B$ be the maximum vertex degree and a subset of vertices in a graph $G$ respectively. In this paper, we study the first (non-trivial) Steklov eigenvalue $\sigma_2$ of $G$ with boundary $B$. Using metrical deformation via…

The k-limited packing number, $L_k(G)$, of a graph $G$, introduced by Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set $X$ of vertices of $G$ such that every vertex of $G$ has at most $k$ elements of $X$ in its…

The $2$-domination number $\gamma_2(G)$ of a graph $G$ is the minimum cardinality of a set $S\subseteq V(G)$ such that every vertex from $V(G)\setminus S$ is adjacent to at least two vertices in $S$. The annihilation number $a(G)$ is the…

Caro, Davila, and Pepper (arXiv:1909.09093) recently proved $\delta(G) \alpha(G)\leq \Delta(G) \mu(G)$ for every graph $G$ with minimum degree $\delta(G)$, maximum degree $\Delta(G)$, independence number $\alpha(G)$, and matching number…

The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. Let $G$ be embeddable on a surface whose Euler characteristic $\chi$ is as large as possible,…

An accessing set in a graph is a subset B of vertices such that there exists D subset of B, such that each vertex of V\B has an even number of neighbors in D. In this paper, we introduce new bounds on the minimal size kappa'(G) of an…

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$. We attach to $N$ two graphs ${\Gamma}_G(N)$ and ${\Gamma}^{\ast}_G(N)$ related to the conjugacy classes of $G$ contained in $N$ and to the set of primes dividing the sizes…

A graph is trivial if it contains one vertex and no edges. The essential connectivity $\kappa^{\prime}$ of $G$ is defined to be the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two non-trivial…

Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if $G$ is an…

Let $G$ be a large-girth $d$-regular graph and $\mu$ be a random process on the vertices of $G$ produced by a randomized local algorithm. We prove the upper bound $(k+1-2k/d)\Bigl(\frac{1}{\sqrt{d-1}}\Bigr)^k$ for the (absolute value of…

A $k$-$\gamma_{c}$-edge critical graph is a graph $G$ with the connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. Further, a $2$-connected graph $G$ is said to be…

For a graph G and an integer t we let mcc_t(G) be the smallest m such that there exists a coloring of the vertices of G by t colors with no monochromatic connected subgraph having more than m vertices. Let F be any nontrivial minor-closed…

The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all $S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number of components of $G-S$. We…

A $k$-fault-tolerant connectivity preserver of a directed $n$-vertex graph $G$ is a subgraph $H$ such that, for any edge set $F \subseteq E(G)$ of size $|F| \le k$, the strongly connected components of $G - F$ and $H - F$ are the same.…