Related papers: Diameter Constraints in 2-distance Graphs

For a graph $G$, the spectral radius of $G$ is the largest eigenvalue of its adjacency matrix. A connected factor of $G$ is a connected spanning subgraph of $G$. For example, a spanning tree of $G$ is a 1-connected factor of $G$. Let $G$ be…

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. An edge subset $F\subseteq E(G)$ is called a restricted edge-cut if $G-F$ is disconnected and has no isolated vertices. The restricted edge-connectivity $\lambda'(G)$ of $G$ is…

The notion of metric dimension, $dim(G)$, of a graph $G$, as well as a number of variants, is now well studied. In this paper, we begin a local analysis of this notion by introducing $cdim_G(v)$, \emph{the connected metric dimension of $G$…

A fractional matching of $G$ is a function $f: E(G)\to [0,1]$ such that $\sum_{e\in E_G(v_i)}f(e)\le 1$ for any $v_i\in V(G)$, where $E_G(v_i)=\{e: e\in E(G) \ \textrm{and}\ e \ \textrm{is incident with} \ v_i\}$. Let $\alpha_f(G)$ denote…

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…

For $E \subset {\Bbb F}_q^d$, $d \ge 2$, where ${\Bbb F}_q$ is the finite field with $q$ elements, we consider the distance graph ${\mathcal G}^{dist}_t(E)$, $t \not=0$, where the vertices are the elements of $E$, and two vertices $x$, $y$…

Let $G$ be a connected graph, and let $b$ and $k$ be two positive integers with $b\equiv1$ (mod 2). A $[1,b]$-odd factor of $G$ is a spanning subgraph $F$ of $G$ with $d_F(v)\equiv1$ (mod 2) and $1\leq d_F(v)\leq b$ for every $v\in V(G)$. A…

For a finite group $G$, let $\mathrm{diam}(G)$ denote the maximum diameter of a connected Cayley graph of $G$. A well-known conjecture of Babai states that $\mathrm{diam}(G)$ is bounded by ${(\log_{2} |G|)}^{O(1)}$ in case $G$ is a…

Brualdi and Hoffman proposed a well-known problem of determining the graph with maximum adjacency spectral radius among all graphs with given size $m$. Early work by Friedland and Stanley addressed some specific cases. This problem was…

Given a finite simple graph $G$, let $\operatorname{M}(G)$ denote its 2-distance graph, in which two vertices are adjacent if and only if they have distance 2 in $G$. In this paper, we consider the periodic behavior of the sequence $G,…

The distinguishing index of a simple graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. It was conjectured by Pil\'sniak (2015) that for any 2-connected…

A graph on $2k$ vertices is path-pairable if for any pairing of the vertices the pairs can be joined by edge-disjoint paths. The so far known families of path-pairable graphs have diameter of length at most 3. In this paper we present an…

The double graph of a graph $G$ is defined as $\mathcal{D}[G]$ = $G \times T_2$, where \(T_2\) is the total graph with 2 vertices and $\times$ stands for the Kronecker product of graphs. In this paper, sufficient conditions for double…

For a graph $G$, $k(G)$ denotes its connectivity. A graph is super connected if every minimum vertex-cut isolates a vertex. Also $k_{1}$-connectivity of a connected graph is the minimum number of vertices whose deletion gives a disconnected…

In this paper, we give a relationship between the covering number of a simple graph $G$, $\beta(G)$, and a new parameter associated to $G$ which is called 2-degree-packing number of $G$, $\nu_2(G)$. We prove that$$\lceil…

Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G = (V,E)$ be a vertex transitive graph, let $A \subseteq V$ be a finite vertex-set with…

The toughness of a graph $G$ is defined as the minimum value of $|S|/c(G-S)$ over all cutsets $S$ of $G$ if $G$ is noncomplete, and is defined to be $\infty$ if $G$ is complete. For a real number $t$, we say that $G$ is $t$-tough if its…

This paper generalizes and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature…

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 $\left\lfloor\frac{2}{3}(d-\lambda_{2}(G))\right\rfloor$…

For a graph $G$, the $k$-recolouring graph $\mathcal{R}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and two colourings are joined by an edge if they differ in colour on exactly one vertex. We prove that for all $n \ge…