Related papers: A Study on Topological Integer Additive Set-Labeli…

Let $\gamma(G)$ denote the domination number of a graph $G$. A vertex $v\in V(G)$ is called a \emph{critical vertex} of $G$ if $\gamma(G-v)=\gamma(G)-1$. A graph is called \emph{vertex-critical} if every vertex of it is critical. In this…

A functor from the category of directed trees with inclusions to the category of commutative C*-algebras with injective *-homomorphisms is constructed. This is used to define a functor from the category of directed graphs with inclusions to…

Let $S=(G, \sigma)$ be a signed graph where $G=(V, E)$ is a graph called the underlying graph of $S$ and $\sigma:E(G) \rightarrow \{+,~-\}$. Let $f:V(G) \rightarrow \{1,2,\dots,|V(G)|\}$ such that $\sigma(uv)=+$ if and only if $f(u)$ and…

Say that an edge of a graph G dominates itself and every other edge adjacent to it. An edge dominating set of a graph G = (V,E) is a subset of edges E' of E which dominates all edges of G. In particular, if every edge of G is dominated by…

A class of graphs is $\chi$-bounded if there is a function $f$ such that $\chi(G)\le f(\omega(G))$ for every induced subgraph $G$ of every graph in the class, where $\chi,\omega$ denote the chromatic number and clique number of $G$…

An automorphism on a graph $G$ is a bijective mapping on the vertex set $V(G)$, which preserves the relation of adjacency between any two vertices of $G$. An automorphism $g$ fixes a vertex $v$ if $g$ maps $v$ onto itself. The stabilizer of…

The automorphisms of a graph act naturally on its set of labeled imbeddings to produce its unlabeled imbeddings. The imbedding sum of a graph is a polynomial that contains useful information about a graph's labeled and unlabeled imbeddings.…

The forgotten topological index of a graph $G$, denoted by $F(G)$, is defined as the sum of weights $d(u)^{2}+d(v)^{2}$ over all edges $uv$ of $G$ , where $d(u)$ denotes the degree of a vertex $u$. In this paper, we give sharp upper bounds…

For graphs $G$ and $H$, a homomorphism from $G$ to $H$ is a function $\varphi \colon V(G) \to V(H)$, which maps vertices adjacent in $G$ to adjacent vertices of $H$. A homomorphism is locally injective if no two vertices with a common…

A graph $H$ is an immersion of a graph $G$ if $H$ can be obtained by some sugraph $G$ after lifting incident edges. We prove that there is a polynomial function $f:\Bbb{N}\times\Bbb{N}\rightarrow\Bbb{N}$, such that if $H$ is a connected…

Given a graph $G$ with $n$ vertices and a bijective labeling of the vertices using the integers $1,2,\ldots, n$, we say $G$ has a peak at vertex $v$ if the degree of $v$ is greater than or equal to 2, and if the label on $v$ is larger than…

For a simple connected graph $G$ of order $n$, a bijective function $f:V(G)\to\{1,2,\cdots,n\}$ is said to be a Legendre cordial labeling modulo $p$, where $p$ is an odd prime, if the induced function $f_p^*:E(G)\to \{0,1\}$, defined by…

We introduce and study the pinnacle sets of a simple graph $G$ with $n$ vertices. Given a bijective vertex labeling $\lambda\,:\,V(G)\rightarrow [n]$, the label $\lambda(v)$ of vertex $v$ is a pinnacle of $(G, \lambda)$ if…

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the…

In this paper, we proposed the \textit{link injection}, a novel method that helps any differentiable graph machine learning models to go beyond observed connections from the input data in an end-to-end learning fashion. It finds out (weak)…

The purpose of this note is to define a graph whose vertex set is a finite group $G$, whose edge set is contained in that of the commuting graph of $G$ and contains the enhanced power graph of $G$. We call this graph the deep commuting…

There is a Turing computable embedding $\Phi$ of directed graphs $A$ in undirected graphs. Moreover, there is a fixed tuple of formulas that give a uniform interpretation; i.e., for all directed graphs $A$, these formulas interpret $A$ in…

A proper labelling of a graph $G$ is a pair $({\pi},c_{\pi})$ in which ${\pi}$ is an assignment of numeric labels to some elements of $G$, and $c_{\pi}$ is a colouring induced by ${\pi}$ through some mathematical function over the set of…

Frank Harary introduced the concept of integral sum graph. A graph $G$ is an \emph{ integral sum graph} if its vertices can be labeled with distinct integers so that $e = uv$ is an edge of $G$ if and only if the sum of the labels on…

Graph neural networks (GNN) have achieved remarkable success in a wide range of tasks by encoding features combined with topology to create effective representations. However, the fundamental problem of understanding and analyzing how graph…