Let G be a nonempty simple graph with vertex set V(G) and an edge E(G). For every injective labeling f:V(G)?Z, there are two induced labelings, namely f+:E(G)?Z defined by f+(uv)=f(u)+f(v), f?:E(G)?Z f?(uv)=|f(u)?f(v)|. The sum index the difference minimum cardinalities of ranges f+ f?, respectively. We provide upper lower bounds on index, determine various families graphs. also interesting con...

Let G be a finite abelian group, written additively, and H subgroup of G. The sum graph $$\varGamma _{G,H}$$ is the with vertex set G, in which two distinct vertices x y are joined if $$x+y\in H{\setminus }\{0\}$$ . These graphs form fairly large class Cayley graphs. Among cases have been considered previously prime graphs, case where $$H=pG$$ for some number p. In this paper we present their s...

let $r$ be a commutative ring with identity. an ideal $i$ of a ring $r$is called an annihilating ideal if there exists $rin rsetminus {0}$ such that $ir=(0)$ and an ideal $i$ of$r$ is called an essential ideal if $i$ has non-zero intersectionwith every other non-zero ideal of $r$. thesum-annihilating essential ideal graph of $r$, denoted by $mathcal{ae}_r$, isa graph whose vertex set is the set...

let $r$ be a commutative ring with zero-divisor set $z(r)$. the total graph of $r$, denoted by$t(gamma(r))$, is the simple (undirected) graph with vertex set  $r$ where two distinct vertices areadjacent if their sum lies in $z(r)$.this work considers minimum zero-sum $k$-flows for $t(gamma(r))$.both for $vert rvert$ even and the case when $vert rvert$ is odd and $z(g)$ is an ideal of $r$it is s...

In this contribution, we first investigate sharp bounds for the reciprocal sum-degree distance of graphs with a given matching number. The corresponding extremal graphs are characterized completely. Then we explore the k-decomposition for the reciprocal sum-degree distance. Finally,we establish formulas for the reciprocal sum-degree distance of join and the Cartesian product of graphs.

Abstract For a simple graph with vertex set { v 1 , 2 … n } \left\{{v}_{1},{v}_{2},\ldots ,{v}_{n}\right\} and degree sequence d i ...