in this paper, we first collect the earlier results about some graph operations and then wepresent applications of these results in working with chemical graphs.

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the annihilating-ideal graph of a commutative ring $r$ is denoted by $ag(r)$, whose vertices are all nonzero ideals of $r$ with nonzero annihilators and two distinct vertices $i$ and $j$ are adjacent if and only if $ij=0$. in this article, we completely characterize rings $r$ when $gr(ag(r))neq 3$.

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.

it is necessary to generate the automorphism group of a chemical graph in computer-aidedstructure elucidation. an euclidean graph associated with a molecule is defined by a weightedgraph with adjacency matrix m = [dij], where for i≠j, dij is the euclidean distance between thenuclei i and j. in this matrix dii can be taken as zero if all the nuclei are equivalent. otherwise,one may introduce dif...

in this paper, we introduce the notions of product vague graph, balanced product vague graph, irregularity and total irregularity of any irregular vague graphs and some results are presented. also, density and balanced irregular vague graphs are discussed and some of their properties are established. finally we give an application of vague digraphs.

Let G(V;E) be a graph. The common neighborhood graph (congraph) of G is a graph with vertex set V , in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of congraphs under graph operations; Graph :::::union:::::, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley graphs and its c...

The Merrifield-Simmons index of a graph is defined as the total number of the independent sets of the graph and the Hosoya index of a graph is defined as the total number of the matchings of the graph. In this paper, we give formula for Merrifield-Simmons and Hosoya indices of some classes of cartesian product of two graphs K{_2}×H, where H is a path graph P{_n}, cyclic graph C{_n}, or star gra...

‎in this paper we give a characterization for all commutative‎ ‎rings with $1$ whose zero-divisor graphs are $c_4$-free.‎

Introduction Indeed, poverty among older persons is still a major problem in many developing countries as well as several developed ones (Glasgow, 1993). It has been widely documented that poverty is negatively associated with health and well-being (Chan et al., 2002; Zimmer, 2008). Hence, the older persons living in poverty may be vulnerable to economic hardships and health problems. Prior stu...