Abstract
In this paper, we studied the very cost effective graph property for the join graph of two graphs. In general this is may or may not be a very cost effective graph. We obtained the conditions for the join graph of two graphs to be a very cost effective graph. First we proved that the join graph Pn∨Pm of path graphs is very cost effective graph if n+m is an even number and is not if n+m is an odd number. Then we proved that the join graph of any two cycle graphs Cn and Cm where n, m are both odd is very cost effective, and the join graph Pn∨Cn is a very cost effective graph if n is an odd number. Also we proved that the join graph G1∨G2 of two very cost effective graphs G1 and G2 is a very cost effective graph if n(G1) + n(G2) is even. Finally we proved that the graph folding of the join graph of two very cost effective graphs not always very cost effective but this will be the case if the sum of the numbers of the vertices in the image of the graph folding is even.
Highlights
A graph G=(V,E) is a nonempty, finite set of elements called vertices together with a set of unordered pairs of distinct vertices of G called edges
This mean that the vertex u is adjacent to the same number of the vertices of S2 and V2\S2, and the join graph Pn ⋁Pm is not a very cost effective graph
Let u ε (V1\S1), u must be adjacent to more vertices in S1 than in V1\S1 certainly by at least two and since u is adjacent to all the vertices of S2 and (V2\S2) and n(S2)=n(V2\S2)−1, u is still a very cost effective vertex in the join graph Cn ⋁Cm
Summary
A graph G=(V,E) is a nonempty, finite set of elements called vertices together with a set of unordered pairs of distinct vertices of G called edges. If e={u, v} is an edge of a graph G, u and v are adjacent vertices, while u and e are incident [1]. The degree (valency) of a vertex v in a graph G is the number of edges incident to v [2]. A graph is said to be connected if every pair of vertices has a path connecting them [8], otherwise is called disconnected. A vertex v in a set S is said to be cost effective if it is adjacent to at least as many vertices in V\S as in S, that is, |N(v) ∩ S| ≤ |N(v) ∩ (V\S)|. A vertex v is very cost effective if it is adjacent to more vertices in V\S than in S, that is, |N(v) ∩ S| < |N(v) ∩ (V\S)|. Graphs that have a (very) cost effective bipartition are called (very) cost effective graphs [7].
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