Tosha-degree equivalence signed graphs

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.