COMMON-EDGE SIGNED GRAPH OF A SIGNED GRAPH
Abstract: A Smarandachely
k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G; σ) (S
= (G; µ)) where G = (V; E) is a graph called underlying graph of S and σ : E !
(e1; e2; :::; ek) (µ : V ! (e1; e2; :::; ek)) is a function, where each ei 2
f+; −g. Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked
graph is abbreviated a signed graph or a marked graph. The commonedge graph of
a graph G = (V; E) is a graph CE(G) = (VE; EE), where VE = fA ⊆ V ; jAj = 3, and A is a connected setg and two vertices in VE
are adjacent ifthey have an edge of G in common. Analogously, one can define
the common-edge signed graph of a signed graph S = (G; σ) as a signed graph
CE(S) = (CE(G); σ0), where CE(G) is the underlying graph of CE(S), where for
any edge (e1e2; e2e3)in CE(S), σ0(e1e2; e2e3) = σ(e1e2)σ(e2e3). It is shown
that for any signed graphS, its common-edge signed graph CE(S) is balanced.
Further, we characterizesigned graphs S for which S ∼ CE(S), S ∼ L(S), S ∼ J(S), CE(S) ∼ L(S)
and CE(S) ∼ J(S), where L(S) and J(S) denotes line
signed graph and jump signed graph of S respectively.
Key words and Phrases: Smarandachely k-signed graphs, Smarandachely k-marked graphs,
balance, switching, common-edge signed graph, line signed graph, jumpsigned
graph
Author: P. Siva Kota Reddy, E.
Sampathkumar and M. S. Subramanya
Journal Code: jpmatematikagg100011
