a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

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in this paper we study the coprime graph of a group $g$. the coprime graph of a group $g$, is a graph whose vertices are elements of $g$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. in this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. also we study the automorphism group of the coprime graph.

In this paper we study a random graph with N nodes, where node j has degree Dj and{Dj}j=1 are i.i.d. with P(Dj ≤ x) = F (x). We assume that 1 − F (x) ≤ cx for some τ > 3and some constant c > 0. This graph model is a variant of the so-called configuration model,and includes heavy tail degrees with finite variance.The minimal number of edges between two arbitrary connected nod...

Exploring the many approaches and issues involved in developing technologies for wellbeing---from including environmental concerns to building long-lasting, transdisciplinary partnerships both inside beyond academy.

A connected graph G is said to be neighbourly irregular graph if no two adjacent vertices of G have same degree. In this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.

the energy of a graph is equal to the sum of the absolute values of its eigenvalues. two graphs of the same order are said to be equienergetic if their energies are equal. we point out the following two open problems for equienergetic graphs. (1) although it is known that there are numerous pairs of equienergetic, non-cospectral trees, it is not known how to systematically construct any such pa...

We propose a new graph metric and study its properties. In contrast to the standard distance in connected graphs [5], it takes into account all paths between vertices. Formally, it is defined as d(i, j) = qii+qjj−qij−qji [11], where qij is the (i, j)-entry of the relative forest accessibility matrix Q(ε) = (I + εL)−1, L is the Laplacian matrix of the (weighted) (multi)graph, and ε is a positive...