Let G : (V, E, ω) be a finite, connected, weighted graph without loops and multiple edges. In a weighted graph each arc is assigned a weight by the weight function ω: E    . A u−v path P in G is called a weighted u−v geodesic if the weighted distance between u and v is calculated along P. The strength of a path is the minimum weight of its arcs, and length of a path is the number of edges in...

Let tt : (V, E, W ) be a finite, connected, weighted graph without loops and multiple edges. In a weighted graph each arc is assigned a weight by the weight function W : E +. A u v path P in tt is called a weighted u v geodesic if the weighted distance between u and v is calculated along P . The strength of a path is the minimum weight of its arcs, and length of a path is the number of edges in...

A set S ⊆ V (G) is called a geodetic if every vertex of G lies on shortest u-v path for some u, v ∈ S, the minimum cardinality among all sets number and denoted by . C chromatic contains vertices different colors in G, geo-chromatic Sc both set. The G. In this paper, we determine 2-cartesian product standard graphs like complete graphs, cycles paths.

The maximum flow algorithm has long been known for calculating the minimum edge cut of any two vertices of a connected graph. The original algorithm, however, does not tell us which edges should be taken exactly and therefore there could be more than one way to construct a minimum-edge-cut set. In this paper, we propose a new method to get a minimum-edge-cut set by selecting edges of a graph in...

We use a new approximation measure, the differential approximation ratio, to derive polynomial-time approximation algorithms for minimum set covering (for both weighted and unweighted cases), minimum graph coloring and bin-packing. We also propose differentialapproximation-ratio preserving reductions linking minimum coloring, minimum vertex covering by cliques, minimum edge covering by cliques ...

For any vertex v and any edge e in a non-trivial connected graph G, the eccentricity e(v) of v is e(v) =max{d(v, u) : u ∈ V }, the vertex-to-edge eccentricity e1(v) of v is e1(v) = max{d(v, e) : e ∈ E}, the edge-to-vertex eccentricity e2(e) of e is e2(e) = max{d(e, u) : u ∈ V } and the edge-to-edge eccentricity e3(e) of e is e3(e) = max{d(e, f) : f ∈ E}. The set C(G) of all vertices v for which...

‎‎let $g=(v,e)$ be a simple graph‎. ‎an edge labeling $f:eto {0,1}$ induces a vertex labeling $f^+:vtoz_2$ defined by $f^+(v)equiv sumlimits_{uvin e} f(uv)pmod{2}$ for each $v in v$‎, ‎where $z_2={0,1}$ is the additive group of order 2‎. ‎for $iin{0,1}$‎, ‎let‎ ‎$e_f(i)=|f^{-1}(i)|$ and $v_f(i)=|(f^+)^{-1}(i)|$‎. ‎a labeling $f$ is called edge-friendly if‎ ‎$|e_f(1)-e_f(0)|le 1$‎. ‎$i_f(g)=v_f(...

The edge C4 graph of a graph G, E4(G) is a graph whose vertices are the edges of G and two vertices in E4(G) are adjacent if the corresponding edges in G are either incident or are opposite edges of some C4. In this paper, we show that there exist infinitely many pairs of non isomorphic graphs whose edge C4 graphs are isomorphic. We study the relationship between the diameter, radius and domina...

The edge set of Kn cannot be decomposed into edge-disjoint hexagons (or 6-cycles) when n =1= 1 or 9 (mod 12). We discuss adding edges to the edge set of Kn so that the resulting graph can be decomposed into edge-disjoint hexagons. This paper gives the solution to this minimum covering of Kn with hexagons problem.