P. Siva Kota Reddy and U. K. Misra

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 19, 2013, Number 4, Pages 86—92

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### Authors and affiliations

P. Siva Kota Reddy

*Department of Mathematics
Siddaganga Institute of Technology
B. H. Road, Tumkur–572 103, India*

U. K. Misra

*Department of Mathematics
Berhampur University
Berhampur–760 007, Orissa, India*

### Abstract

A signed graph (marked graph) is an ordered pair S = (G; σ) (S = (G; μ)), where G = (V, E) is a graph called the underlying graph of S and σ : E → {+, −} (μ : V → {+, −}) is a function. The restricted super line graph of index r of a graph G, denoted by RL_{r}(G). The vertices of RLr(G) are the r-subsets of E(G) and two vertices P = {p_{1}, p_{2} …, p_{r}} and Q = {q_{1}, q_{2} …, q_{r}} are adjacent if there exists exactly one pair of edges, say p_{i} and q_{j} , where 1 ≤ i; j ≤ r, that are adjacent edges in G.

Analogously, one can define the restricted super line signed graph of index r of a signed graph S = (G; σ) as a signed graph RLr(S) = (RLr(G); σ′), where RLr(G) is the underlying graph of RLr(S), where for any edge PQ in RLr(S), σ′(PQ) = σ(P)σ(Q). It is shown that for any signed graph S, its RLr(S) is balanced and we offer a structural characterization of restricted super line signed graphs of index r.

Further, we characterize signed graphs S for which RLr(S) ~ Lr(S) and RLr(S) ≅ Lr(S), where ~ and ≅ denote switching equivalence and isomorphism and RLr(S) and Lr(S) are denotes the restricted super line signed graph of index r and super line signed graph of index r of S, respectively.

### Keywords

- Signed graphs
- Marked graphs
- Balance
- Switching
- Restricted super line signed graph
- Super line signed graphs
- Negation

### AMS Classification

- 05C22

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## Cite this paper

Siva Kota Reddy, P. & Misra, U. (2013). Restricted super line signed graph *RL _{r}*(

*S*), Notes on Number Theory and Discrete Mathematics, 19(4), 86-92.