vspace{0.2cm} Let $G$ be a graph and $f:V(G)rightarrow {1,2,3,.....left|V(G)right|}$ be a bijection. Let $p_{uv}=f(u)f(v)$ and\ $ d_{uv}= begin{cases} left[frac{f(u)}{f(v)}right] ~~if~~ f(u) geq f(v)\ \ left[frac{f(v)}{f(u)}right] ~~if~~ f(v) geq f(u)\ end{cases} $\ for all edge $uv in E(G)$. For each edge $uv$ assign the label $1$ if $gcd (p_{u...

This paper proposes two models of adding relations to a linking pin type organization structure where every pair of siblings in a complete Kary tree of height H is adjacent: (i) a model of adding an edge between two nodes with the same depth N and (ii) a model of adding edges between every pair of nodes with the same depth N . For each of the two models, an optimal depth N∗ is obtained by maxim...

let $g$ be a connected graph of order $3$ or more and $c:e(g)rightarrowmathbb{z}_k$‎ ‎($kge 2$) a $k$-edge coloring of $g$ where adjacent edges may be colored the same‎. ‎the color sum $s(v)$ of a vertex $v$ of $g$ is the sum in $mathbb{z}_k$ of the colors of the edges incident with $v.$ the $k$-edge coloring $c$ is a modular $k$-edge coloring of $g$ if $s(u)ne s(v)$ in $mathbb{z}_k$ for all pa...

Abstract—This paper proposes a model of adding relation to a linking pin organization structure where every pair of siblings in a complete K-ary tree of height H is adjacent such that the communication of information in the organization becomes the most efficient. For a model of adding an edge between a node with a depth M and its descendant with a depth N, we formulated the total shortening di...

This paper proposes two models of adding relations to a linking pin type organization structure where every pair of siblings in a complete Kary tree of height H is adjacent: (i) a model of adding an edge between two nodes with the same depth N and (ii) a model of adding edges between every pair of nodes with the same depth N . For each of the two models, an optimal depth N∗ is obtained by maxim...

For any integer k > 0, a tree T is k-cordial if there exists a labeling of the vertices of T by Zk, inducing edge-weights as the sum modulo k of the labels on incident vertices to a given edge, which furthermore satisfies the following conditions: 1. Each label appears on at most one more vertex than any other label. 2. Each edge-weight appears on at most one more edge than any other edge-weigh...

A graph is called supermagic if there is a labeling of edges where the edges are labeled with consecutive distinct positive integers such that the sum of the labels of all edges incident with any vertex is constant. A graph G is called degree-magic if there is a labeling of the edges by integers 1, 2, ..., |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal t...

Let $G$ be a graph. Let $f:V(G)to{0,1,2, ldots, k-1}$ be a map where $k in mathbb{N}$ and $k>1$. For each edge $uv$, assign the label $left|f(u)-f(v)right|$. $f$ is called a $k$-total difference cordial labeling of $G$ if $left|t_{df}(i)-t_{df}(j)right|leq 1$, $i,j in {0,1,2, ldots, k-1}$ where $t_{df}(x)$ denotes the total number of vertices and the edges labeled with $x$.A graph with admits a...