Related papers: A Study on the Modular Sumset Labeling of Graphs

Given an edge labeling $f$ of a graph $G$, a vertex $v$ is called an $AR$-vertex, if $v$ has distinct edge weight sums for each distinct subset of edges incident on $v$. An injective edge labeling $f$ of a graph $G$ is called an…

Let $G$ be a graph, a dominating induced matching (DIM) of $G$ is an induced matching that dominates every edge of $G$. In this paper we show that if a graph $G$ has a DIM, then $\chi(G) \leqslant 3$. Also, it is shown that if $G$ is a…

Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple graph in which a non-zero submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper submodule $K$…

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph which was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced. In this paper we study the F-index of four…

Various algorithms have been proposed to enumerate all connected induced subgraphs of a graph $G=(V,E)$. As a variation we enumerate all "packings of connected sets", i.e. partitions $\Pi$ of $V$ with the property that each part of $\Pi$…

This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II…

Given a graph $G$, a subset $M$ of $V(G)$ is a module of $G$ if for each $v\in V(G)\setminus M$, $v$ is adjacent to all the elements of $M$ or to none of them. For instance, $V(G)$, $\emptyset$ and $\{v\}$ ($v\in V(G)$) are modules of $G$…

A set $D \subseteq V(G)$ is a \emph{total dominating set} of $G$ if for every vertex $v \in V(G)$ there exists a vertex $u \in D$ such that $u$ and $v$ are adjacent. A total dominating set of $G$ of minimum cardinality is called a…

Motivated in part by combinatorial applications to certain sum-product phenomena, we introduce unimodular graphs over finite fields and, more generally, over finite valuation rings. We compute the spectrum of the unimodular graphs, by using…

Let $V$ be a left $R$-module where $R$ is a (not necessarily commutative) ring with unit. The intersection graph $\cG(V)$ of proper $R$-submodules of $V$ is an undirected graph without loops and multiple edges defined as follows: the vertex…

Consider a finite and simple graph $G=(V,E)$ with maximum degree $\Delta$. A strong Roman dominating function over the graph $G$ is understood as a map $f : V (G)\rightarrow \{0, 1,\ldots , \left\lceil \frac{\Delta}{2}\right\rceil+ 1\}$…

The injective polynomial modules for a general linear group $G$ of degree $n$ are labelled by the partitions with at most $n$ parts. Working over an algebraically closed field of characteristic $p$, we consider the question of which…

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…

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…

To a simple graph we associate a so-called graph series, which can be viewed as the Hilbert--Poincar\'e series of a certain infinite jet scheme. We study new $q$-representations and examine modular properties of several examples including…

A modulator of a graph G to a specified graph class H is a set of vertices whose deletion puts G into H. The cardinality of a modulator to various tractable graph classes has long been used as a structural parameter which can be exploited…

Let $G$ be a graph that admits a perfect matching. A {\sf forcing set} for a perfect matching $M$ of $G$ is a subset $S$ of $M$, such that $S$ is contained in no other perfect matching of $G$. This notion originally arose in chemistry in…

We extend the concept of the law of a finite graph to graphings, which are, in general, infinite graphs whose vertices are equipped with the structure of a probability space. By doing this, we obtain a vast array of new unimodular measures.…

Given a simple graph $G$, a dominating set in $G$ is a set of vertices $S$ such that every vertex not in $S$ has a neighbor in $S$. Denote the domination number, which is the size of any minimum dominating set of $G$, by $\gamma(G)$. For…

We settle affirmatively a conjecture posed in [S. M. Hegde, Set colorings of graphs, European Journal of Combinatorics 30 (4) (2009), 986--995]: If some subsets of a set X are assigned injectively to all vertices of a complete bipartite…