A Study on Edge-Set Graphs of Certain Graphs

Abstract

Let G(V, E) simple connected graph, with |E| = . In this paper, we define an edge-set graph G G constructed from the graph G such that any vertex v s,i of G G corresponds to the i-th s-element subset of E(G) and any two vertices v s,i , v k,m of G G are adjacent if and only if there is at least one edge in the edge-subset corresponding to v s,i which is adjacent to at least one edge in the edge-subset corresponding to v k,m , where s, k are positive integers. It can be noted that the edge-set graph G G of a graph G is dependent on both the structure of G as well as the number of edges . We also discuss the characteristics and properties of the edge-set graphs corresponding to certain standard graphs.

Theb-chromatic number of certain graphs and digraphs

“…Recently, the families of graphs namely, Jaco graphs, linear Jaco graphs, ornated graphs, Rasta graphs, set-graphs, edge-set graphs and edge-joint graphs were introduced in different studies in [6,7,8,10,11,12].…”

“…Another class of graphs similar to the set-graphs is the class of edge-set graphs which has been introduced in [12] as follows. (i) |V (G G )| = 2 − 1 so that there exists a one to one correspondence between V (G G ) and E.…”

“…Clearly, the edge-set graph is dependent on both the number of edges and the structure of G. It is known that ϕ(G) ≤ ∆(G) + 1 (see [5,13]). In [12] it is proved that the edge-set of G on n vertices is a complete graph if and only if G is a star graph. Hence, it can be said that ϕ(G G ) ≤ ϕ(K n ).…”

“…[12] Let G(V, E) be a non-empty finite graph with |E| = and E = P(E) − {∅}, where P(E) is the power set of the edge set E(G). For 1 ≤ s ≤ , let S be the collection of all s-element subsets of E(G) and E s,i be the i-th element of S. Then the edge-set graph corresponding to G, denoted by G E ( ) or G G , is the graph with the following properties.…”

On the Pythagorean Holes of Certain Graphs

“…The primitive hole number of the underlying graph of a Jaco graph has been determined in the following theorem (see [6]). [6] Let J * n (1) be the underlying graph of a finite Jaco Graph J n (1) with Jaconian vertex v i , where n is a positive integer greater than or equal to 4.…”