Related papers: Edge distribution in generalized graph products

For a graph G, its rth power G^r has the same vertex set as G, and has an edge between any two vertices within distance r of each other in G. We give a lower bound for the number of edges in the rth power of G in terms of the order of G and…

The graph G=(V,E) is called Konig-Egervary if the sum of its independence number and its matching number equals its order. Let RV(G) denote the number of vertices v such that G-v is Konig-Egervary, and let RE(G) denote the number of edges e…

Researchers developing implementations of distributed graph analytic algorithms require graph generators that yield graphs sharing the challenging characteristics of real-world graphs (small-world, scale-free, heavy-tailed degree…

Let $d,n\in \mathbb{N}$ be such that $d=\omega(1)$, and $d\le n^{1-a}$ for some constant $a>0$. Consider a $d$-regular graph $G=(V, E)$ and the random graph process that starts with the empty graph $G(0)$ and at each step $G(i)$ is obtained…

For a graph $G$, the set of vertices in $G$ is denoted by $V(G)$, and the set of edges in $G$ is denoted by $E(G)$. A fractional $[a,b]$-factor of a graph $G$ is a function $h$ from $E(G)$ to $[0,1]$ satisfying $a\leq d_G^{h}(v)\leq b$ for…

Let $G$ be a graph, $S$ be a set of vertices of $G$, and $\lambda(S)$ be the maximum number $\ell$ of pairwise edge-disjoint trees $T_1, T_2,..., T_{\ell}$ in $G$ such that $S\subseteq V(T_i)$ for every $1\leq i\leq \ell$. The generalized…

Let $G=(V,E)$ be a graph and $A$ its adjacency matrix. We say that a vertex $y \in V$ is a function of vertices $x_1, \ldots, x_k \in V$ if there exists a Boolean function $f$ of $k$ variables such that for any vertex $z \in V - \{y, x_1,…

An injective coloring of a given graph G = (V, E) is a vertex coloring of G such that any two vertices with common neighbor receive distinct colors. An e-injective coloring of a graph G is a vertex coloring of G such that any two vertices…

Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this result, this paper systematically…

We study the edge-averaging process on a finite, connected graph $G = (V, E)$. Initially, the vertices in $V$ are endowed with i.i.d.\ real-valued opinions $(f_0(v))_{v \in V}$. Edges are activated according to i.i.d.\ Poisson clocks of…

Computing efficiently a robust measure of similarity or dissimilarity between graphs is a major challenge in Pattern Recognition. The Graph Edit Distance (GED) is a flexible measure of dissimilarity between graphs which arises in…

Let $k$ be a positive integer and let $G$ be a graph with vertex set $V(G)$. A subset $D \subseteq V(G)$ is a $k$-dominating set if every vertex outside $D$ is adjacent to at least $k$ vertices in $D$. The $k$-domination number…

In 1970 Lov{\'a}sz gave a necessary and sufficient condition for the existence of a factor $F$ in a graph $G$ such that for each vertex $v$, $g(v)\le d_F(v)\le f(v)$, where $g$ and $f$ are two integer-valued functions on $V(G)$ with $g\le…

The \emph{Antimagic Graph Conjecture} asserts that every connected graph $G = (V, E)$ except $K_2$ admits an edge labeling such that each label $1, 2, ..., |E|$ is used exactly once and the sums of the labels on all edges incident with a…

Let the Andr\'{a}sfai graph $\mathrm{And}_k$ be defined as the graph with vertex set $\{v_0,v_1,...c, v_{3k-2}\}$ and two vertices $v_i$ and $v_j$ being adjacent iff $|i-j| \equiv 1\mod 3$. The graphs $\mathrm{And}_k$ are maximal…

Let $G$ be a connected edge-weighted graph of order $n$ and size $m$. Let $w:E(G)\rightarrow \mathbb{R}^{\geq 0}$ be the weighting function. We assume that $w$ is normalised, that is, $\sum_{e\in E(G)} w(e)=m$. The weighted distance…

Given a graph $G$, a subset $S \subseteq V(G)$ is \textit{cycle convex}, if for any vertex $v \in V(G) \setminus S$, the induced subgraph, $G[S \cup \{v\}]$ cannot form a cycle containing the vertex $v$. The \textit{exchange number} of $G$,…

Graph Convolutional Networks (GCNs) have been widely studied. The core of GCNs is the definition of convolution operators on graphs. However, existing Graph Convolution (GC) operators are mainly defined on adjacency matrix and node features…

A graph $G$ is called edge-magic if there is a bijective function $f$ from the set of vertices and edges to the set $\{1,2,\ldots,|V(G)|+|E(G)|\}$ such that the sum $f(x)+f(xy)+f(y)$ for any $xy$ in $E(G)$ is constant. Such a function is…

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized $k$-edge-connectivity $\lambda_k(G)$. For general $k$, the generalized…