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 signed graph (marked graph) is an ordered pair $s=(g,sigma)$$(s=(g,mu))$, where $g=(v,e)$ is a graph called the underlyinggraph of $s$ and $sigma:erightarrow{+,-}$$(mu:vrightarrow{+,-})$ is a function. for a graph $g$, $v(g),e(g)$ and $c(g)$ denote its vertex set, edge set and cut-vertexset, respectively. the lict graph $l_{c}(g)$ of a graph $g=(v,e)$is defined as the graph having vertex set ...

For a simple path Pr on r vertices, the square of Pr is the graph P 2 r on the same set of vertices of Pr, and where every pair of vertices of distance 2 or less in Pr are connected by an edge. Given a (p, q)-graph G with p vertices and q edges, and an integer k ≥ 0, G is said to be k-edge-graceful if the edges can be labeled by k, k+1, . . . , k+q−1 so that the induced vertex sums (mod p) are ...

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 ...

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...

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...

An edge-coloring of a graph G with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of G are distinct and the sum of the colors of the edges of G is minimum. The edge-chromatic sum of a graph G is the sum of the colors of edges in a sum edge-coloring of G. It is known that the problem of finding the edge-chromatic sum of an r-regular (r ≥ 3) graph is N...

Let G be a (p, q) graph. An injective map f : V (G) → {±1,±2, · · · ,± p} iscalled a pair sum labeling if the induced edge function, fe : E(G) → Z − {0} defined byfe(uv) = f(u) + f(v) is one-one and fe(E(G)) is either of the form {± k1,± k2, · · · ,± k q2}or {± k1,± k2, . . . ,± k q−12} ∪ {k q+12} according as q is even or odd. Here we study aboutthe pair...

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H . In this paper, we give a new sufficient condition for a weighted graph to...

let $g$ be a simple graph with vertex set $v(g) = {v_1, v_2,ldots, v_n}$ and edge set $e(g) = {e_1, e_2,ldots , e_m}$. similar tothe randi'c matrix, here we introduce the randi'c incidence matrixof a graph $g$, denoted by $i_r(g)$, which is defined as the$ntimes m$ matrix whose $(i, j)$-entry is $(d_i)^{-frac{1}{2}}$ if$v_i$ is incident to $e_j$ and $0$ otherwise. naturally, therandi'c incidenc...