Related papers: On solid density of Cayley digraphs on finite Abel…

A Cayley graph of a group $H$ is a finite simple graph $\Gamma$ such that its automorphism group ${\rm Aut}(\Gamma)$ contains a subgroup isomorphic to $H$ acting regularly on $V(\Gamma)$, while a Haar graph of $H$ is a finite simple…

A $\Gamma$\emph{-distance magic labeling} of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$, for which there exists $\mu \in \Gamma$, such that the weight $w(x) =\sum_{y\in…

The standard double cover of a graph $\Gamma$ is the direct product $\Gamma\times K_2$. A graph $\Gamma$ is said to be stable if all the automorphisms of $\Gamma\times K_2$ come from its factors. Although the study of stability has…

For a graph $\Gamma$, the multiplicity of the eigenvalue $0$, denoted by $\eta(\Gamma)$, is called the nullity of $\Gamma$. Also the energy of $\Gamma$, denoted by $\mathcal{E}(\Gamma)$, is defined as the sum of the absolute values of the…

Let $G$ be a finite group, and $S$ be a subset of $G\setminus\{1\}$ such that $S=S^{-1}$. Suppose that $Cay(G,S)$ is the Cayley graph on $G$ with respect to the set $S$ which is the graph whose vertex set is $G$ and two vertices $a,b\in G$…

Let $\mathbf{\Gamma} = (V,E)$ be a (non-trivial) finite graph with $\lambda: E \rightarrow \mathbb{R}_{+}$, an edge labelling of $\mathbf{\Gamma}$. Let $\rho : V\rightarrow \mathbb{R}^{2}$ be a map which preserves the edge labelling. The…

Let $G$ be a finite abelian group written additively with identity $0$, and $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin \Omega$. We say that $ \Omega $ has the property \lq\lq{}$us$\rq\rq{} (unique summation),…

Given a weighted graph $G=(V,E,w)$, a partition of $V$ is $\Delta$-bounded if the diameter of each cluster is bounded by $\Delta$. A distribution over $\Delta$-bounded partitions is a $\beta$-padded decomposition if every ball of radius…

Abelian Cayley digraphs can be constructed by using a generalization to $Z^n$ of the concept of congruence in $Z$. Here we use this approach to present a family of such digraphs, which, for every fixed value of the degree, have…

Let $\Gamma$ be a Cayley graph of the permutation group generated by a transposition tree $T$ on $n$ vertices. In an oft-cited paper \cite{Akers:Krishnamurthy:1989} (see also \cite{Hahn:Sabidussi:1997}), it is shown that the diameter of the…

We show that any connected Cayley graph $\Gamma$ on an Abelian group of order $2n$ and degree $\tilde{\Omega}(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $\Gamma$ is…

We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group $G$ has edge density close to the expected one on every induced subgraph of size at least $\log^c |G|$, for any fixed $c > 1$ and $|G|$…

We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing…

An integer distance digraph is the Cayley graph $\Gamma(\mathbb{Z},S)$ of the additive group $\mathbb{Z}$ of all integers with respect to some finite subset $S \subseteq \mathbb{Z}$. The domination ratio of $\Gamma(\mathbb{Z},S)$ is the…

Let $\overrightarrow{G}$ be a directed graph of order $n$ with no component of order less than $4$, and let $\Gamma$ be a finite Abelian group such that $|\Gamma|\geq n+6$. We show that there exists a mapping $\psi$ from the arc set…

Let $S\subset\text{GL}_n(\mathbb Z)$ be a finite symmetric set. We show that if the Zariski closure of $\Gamma=\langle S\rangle$ is a product of $\text{SL}_d$ or a special affine linear group, then the diameter of the Cayley graph…

We show that for certain integers $n$, the problem of whether or not a Cayley digraph $\Gamma$ of $\mathbb Z_n$ is also isomorphic to a Cayley digraph of some other abelian group $G$ of order $n$ reduces to the question of whether or not a…

In this paper, we introduce the concept of $k$-integral graphs. A graph $\Gamma$ is called $k$-integral if the extension degree of the splitting field of the characteristic polynomial of $\Gamma$ over rational field $\mathbb Q$ is equal to…

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $\ell$ from $V$ to an Abelian group $\Gamma$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}\ell(y)$ of every vertex $x \in V$ is equal to the…

A $\Gamma$-distance magic labeling of a graph $G=(V,E)$ with $|V | = n$ is a bijection $f$ from $V$ to an Abelian group $\Gamma$ of order $n$ such that the weight $w(x)=\sum_{y\in N_G(x)}f(y)$ of every vertex $x \in V$ is equal to the same…