# Neighborhood Signed Graphs

Rangarajan, R. (2012) Neighborhood Signed Graphs. Southeast Asian Bulletin of Mathematics, 36 (3). pp. 389-398.

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## Abstract

A \emph{signed graph (marked graph)} is an ordered pair $S = (G, \sigma)$ ($S =(G, \mu)$), where $G=(V,E)$ is a graph called the \emph{underlying graph} of $S$ and $\sigma : E \rightarrow \{+,-\}$ ($\mu : V \rightarrow \{+,-\}$) is a function. The \emph{neighborhood graph} of a graph $G=(V,E)$, denoted by $N(G)$, is a graph on the same vertex set $V$, where two vertices in $N(G)$ are adjacent if, and only if, they have a common neighbor. Analogously, one can define the \emph{neighborhood signed graph} $N(S)$ of a signed graph $S=(G, \sigma)$ as a signed graph, $N(S)=(N(G), \sigma')$, where $N(G)$ is the underlying graph of $N(S)$, and for any edge $e = uv$ in $N(S)$, $\sigma'(e)=\mu(u)\mu(v)$, where for any $v \in V$, $\mu(v) =\prod_{u\in N(v)}\sigma(uv)$. In this paper, we characterize signed graphs $S$ for which $S \sim N(S)$, $S^c \sim N(S)$ and $N(S)\sim J(S)$, where $J(S)$ and $S^c$ denotes jump signed graph and complement of signed graph of $S$ respectively.

Item Type: Article E Mathematical Science > Mathematics Department of > Mathematics C Swapna Library Assistant 27 Aug 2019 06:28 27 Aug 2019 06:28 http://eprints.uni-mysore.ac.in/id/eprint/7114

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