Let G be a finite simple graph with n vertices. The degree of a vertex v, which we write as deg v, is the number of vertices of G adjacent to v. Since there can be at most one edge between any pair of vertices in a simple graph, deg v ≤ n− 1 for each vertex v. One of the most basic results in Graph Theory, which is also easy to prove, is that if we sum the degrees of vertices of a finite simple...

let g be a (p, q) graph. let f : v (g) → {1, 2, . . . , k} be a map. for each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of g if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

let $g$ be a simple connected graph. the edge-wiener index $w_e(g)$ is the sum of all distances between edges in $g$, whereas the hyper edge-wiener index $ww_e(g)$ is defined as {footnotesize $w{w_e}(g) = {frac{1}{2}}{w_e}(g) + {frac{1}{2}} {w_e^{2}}(g)$}, where {footnotesize $ {w_e^{2}}(g)=sumlimits_{left{ {f,g} right}subseteq e(g)} {d_e^2(f,g)}$}. in this paper, we present explicit formula fo...

For a graph $G$ with edge set $E(G)$, the multiplicative sum Zagreb index of $G$ is defined as$Pi^*(G)=Pi_{uvin E(G)}[d_G(u)+d_G(v)]$, where $d_G(v)$ is the degree of vertex $v$ in $G$.In this paper, we first introduce some graph transformations that decreasethis index. In application, we identify the fourteen class of trees, with the first through fourteenth smallest multiplicative sum Zagreb ...

This study considers the addition of relation to an organization structure such that the communication of information between every member in the organization becomes the most efficient. 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. When a new edge between the root and a node ...

This paper proposes a model of adding relation to a linking pin organization structure where every pair of siblings in a complete binary 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 distance ...

The edge Szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph G, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. In this paper, the edge Szeged index of one-pentagonal carbon nanocone CNC5[n] is computed for the first ...

A graph G = (V,E), where V is the set of vertices, and E is the set of edges. An edge e ∈ E is an unordered pair (u,v) in undirected graphs, where u,v∈V . In directed graphs, an edge e is an ordered pair. A path from a vertex u to a vertex v is a sequence of vertices (w0,w1, ...wk), where u = w0, v = wk and (wi−1,wi) ∈ E for all 1 ≤ i ≤ k. The path is a cycle if u = v. The length of a path in a...