# (2, d)-Sigraphs

E. Sampathkumar and M. A. Sriraj
Notes on Number Theory and Discrete Mathematics, ISSN 1310–5132
Volume 19, 2013, Number 4, Pages 16–27
Full paper (PDF, 238 Kb)

## Details

### Authors and affiliations

E. Sampathkumar
Department of Studies in Mathematics, University of Mysore
Mysore–570 006, India

M. A. Sriraj
Department of Mathematics, Vidyavardhaka College of Engineering
P.B. No.206, Gokulam III Stage
Mysore–570 002, India

### Abstract

An edge uv in a graph G is directionally labeled by an ordered pair ab if the label ℓ(uv) on uv is ab in the direction from u to v; and ℓ(vu) = ba. A (2, d)-sigraph G = (V, E) is a graph in which every edge is directionally labeled by an ordered pair ab ∈ {++, −−, +−, −+}. A (2, d)-sigraph G has a uniform-directional edge labeling (ude-labeling) at a vertex u in G, if for each neighbor v of u, either ℓ(vu) ∈ {++, +−} or ℓ(vu) ∈ {−−, −+}. Further, G is ude-balanced if it has such a labeling at each of its vertex. Two characterizations of ude-balanced (2, d)-sigraphs are obtained. Using a notion of 2-splitting of a (2, d)-sigraphs, we define a 2-balanced (2, d)-sigraph, and obtain a characterization of 2-balanced (2, d)-sigraph which is similar to a characterization of balanced sigraphs. Further, the notion of clusterability of signed graphs is extended to (2, d)-sigraphs, and a characterization of clusterable (2, d)-sigraph is obtained. The notions of ude-balance and clusterability are extended to (n, d)-sigraphs. Some applications of (2, d)-sigraphs are also mentioned.

### Keywords

• Signed graph
• (2, d)-sigraph
• Uniform directional labeling
• Clusterability
• Bidirected graph

• 05C22
• 05B20

### References

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## Cite this paper

Sampathkumar, E., & Sriraj, M. A.(2013). (2, d)-Sigraphs. Notes on Number Theory and Discrete Mathematics, 19(4), 16-27.