# Vertex Equitable Labeling of Signed Bistars

- 49 Downloads

## Abstract

A *signed graph* \(S=(S^u,\sigma )\) has a graph \(S^u\) and a function \(\sigma :E(S^u)\longrightarrow \{+,-\}\).
Let *S* has *q* edges and \({\mathcal {A}} = \{0,1,2,\ldots , \lceil \frac{q}{2}\rceil \}\). A vertex labeling \(f:V(S)\longrightarrow {\mathcal {A}}\) which is onto, is said to be a vertex equitable labeling of *S* if it induces a bijective edge labeling \(f^*:E(S)\longrightarrow \{1,2,\ldots ,{\mathfrak {m}},-1,-2,\ldots ,-{\mathfrak {n}}\}\) defined by \(f^*(uv) = \sigma (uv)(f(u)+f(v))\) such that \(|v_f(a)-v_f(b)|\le 1\), \(\forall a, b\in {\mathcal {A}}\), where \(v_f(a)\) is the number of vertices with \(f(v) = a\) and \({\mathfrak {m}}, {\mathfrak {n}}\) are number of positive and negative edges respectively in *S*. A signed graph *S* is said to be vertex equitable if it admits a vertex equitable labeling. In this paper, we study vertex equitable behavior of signed bistars.

## Keywords

Vertex equitable labeling Vertex equitable signed graph Bistar## References

- 1.Gallian JA (2011) A dynamic survey of graph labeling. Electron J Comb 18:1Google Scholar
- 2.Harary F (1957) Structural duality. Behav Sci 2(4):255–265MathSciNetCrossRefGoogle Scholar
- 3.Seenivasan M, Lourdusamy A (2008) Vertex equitable labeling of graphs. J Discret Math Sci Cryptogr 11(6):727–735MathSciNetCrossRefzbMATHGoogle Scholar
- 4.Acharya M, Jain R, Kansal S (2017) Vertex equitable labeling of signed graphs. Electron Notes Discret Math 63:461–468MathSciNetCrossRefzbMATHGoogle Scholar