Tosha-Degree Equivalence Signed Graphs

Rajendra, R. , Reddy, P. S. K.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 2.

Abstract:
The Tosha-degree of an edge \(\alpha \) in a graph \(\Gamma\) without multiple edges, denoted by \(T(\alpha)\), is the number of edges adjacent to \(\alpha\) in \(\Gamma\), with self-loops counted twice. A signed graph (marked graph) is an ordered pair \(\Sigma=(\Gamma,\sigma)\) (\(\Sigma =(\Gamma, \mu)\)), where \(\Gamma=(V,E)\) is a graph called the underlying graph of \(\Sigma\) and \(\sigma : E \rightarrow \{+,-\}\) (\(\mu : V \rightarrow \{+,-\}\)) is a function. In this paper, we define the Tosha-degree equivalence signed graph of a given signed graph and offer a switching equivalence characterization of signed graphs that are switching equivalent to Tosha-degree equivalence signed graphs and \( k^{th}\) iterated Tosha-degree equivalence signed graphs. It is shown that for any signed graph \(\Sigma\), its Tosha-degree equivalence signed graph \(T(\Sigma)\) is balanced and we offer a structural characterization of Tosha-degree equivalence signed graphs.

Keywords: signed graphs, balance, switching, Tosha-degree of an edge, Tosha-degree equivalence signed graph, negation.

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