let g be a (p,q) graph. an injective map f : e(g) → {±1,±2,...,±q} is said to be an edge pair sum labeling if the induced vertex function f*: v (g) → z - {0} defined by f*(v) = σp∈ev f (e) is one-one where ev denotes the set of edges in g that are incident with a vertex v and f*(v (g)) is either of the form {±k1,±k2,...,±kp/2} or {±k1,±k2,...,±k(p-1)/2} u {±k(p+1)/2} according a...

Over 100 distributions, and another 150 synonyms.

As highly discriminant distance-based topological indices, the Balaban index and the sum-Balaban index of a graph $G$ are defined as $J(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=sumlimits_{vin V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the n...

Background & Aim: Eucalyptus L'Her. [an aromatic genera of the family (Myrtaceae)] is native to Australia and other tropical species is planted worldwide. The purpose of this study is to extract and identify the essential oils components of four species of Eucalyptus L'Her. cultivated in botanical garden of Kashan, Central Iran. Experimental: Leaves of E. largiflorens</em...

Proof. Let n = |V |, and assume without loss of generality that |E| = n (else we could remove edges from G until |E| = n and apply the argument). Also assume WLOG that V ∩ E = ∅ (else, we can rename the vertices). Note that we must have n ≥ 3, since a simple graph with 2 or fewer vertices can have at most 1 edge. Now we seek a bijection f : V → E with each v ∈ V satisfying v / ∈ f(v). Construct...

The cartesian product of two hamiltonian graphs is always hamiltonian . For directed graphs, the analogous statement is false . We show that the cartesian product C, x C„ 2 of directed cycles is hamiltonian if and only if the greatest common divisor (g .c .d .) d of n, and n 2 is at least two and there exist positive integers d,, d 2 so that d, + d2 = d and g.c.d . (n,, d,) = g .c.d. (n 2, d2 )...

In an undirected graph G = (V,E) with a weight function w : E×V → Q+, the weighted degree dw(v;E) of a vertex v is defined as ∑ {w(e, v) | e ∈ E incident with v}. In this paper, we consider a network design problem with upper-bound on weighted degree of each vertex. Inputs of the problem are an undirected graph G = (V,E) with E = E1 ∪̇ E2 ∪̇ E3, weights w1 : E1 × V → Q+, μ : E2 → Q+, ν : E3 → Q+,...