Document Type : Research Paper

**Authors**

DELHI TECHNOLOGICAL UNIVERSITY, DELHI - INDIA

**Abstract**

A *signed graph* (or, in short, *sigraph*) $S=(S^u,\sigma)$ consists of an underlying graph $S^u :=G=(V,E)$ and a function $\sigma:E(S^u)\longrightarrow \{+,-\}$, called the signature of $S$. A *marking* of $S$ is a function $\mu:V(S)\longrightarrow \{+,-\}$. The *canonical marking* of a signed graph $S$, denoted $\mu_\sigma$, is given as $$\mu_\sigma(v) := \prod_{vw\in E(S)}\sigma(vw).$$

The*line graph* of a graph $G$, denoted $L(G)$, is the graph in which edges of $G$ are represented as vertices, two of these vertices are adjacent if the corresponding edges are adjacent in $G$. There are three notions of a *line signed graph* of a signed graph $S=(S^u,\sigma)$ in the literature, viz., $L(S)$, $L_\times(S)$ and $L_\bullet(S)$, all of which have $L(S^u)$ as their underlying graph; only the rule to assign signs to the edges of $L(S^u)$ differ. Every edge $ee'$ in $L(S)$ is negative whenever both the adjacent edges $e$ and $e'$ in S are negative, an edge $ee'$ in $L_\times(S)$ has the product $\sigma(e)\sigma(e')$ as its sign and an edge $ee'$ in $L_\bullet(S)$ has $\mu_\sigma(v)$ as its sign, where $v\in V(S)$ is a common vertex of edges $e$ and $e'$.

The line-cut graph (or, in short,*lict graph*) of a graph $G=(V,E)$, denoted by $L_c(G)$, is the graph with vertex set $E(G)\cup C(G)$, where $C(G)$ is the set of cut-vertices of $G$, in which two vertices are adjacent if and only if they correspond to adjacent edges of $G$ or one vertex corresponds to an edge $e$ of $G$ and the other vertex corresponds to a cut-vertex $c$ of $G$ such that $e$ is incident with $c$.

In this paper, we introduce*dot-lict signed graph* (or $\bullet$*-lict signed graph*} $L_{\bullet_c}(S)$, which has $L_c(S^u)$ as its underlying graph. Every edge $uv$ in $L_{\bullet_c}(S)$ has the sign $\mu_\sigma(p)$, if $u, v \in E(S)$ and $p\in V(S)$ is a common vertex of these edges, and it has the sign $\mu_\sigma(v)$, if $u\in E(S)$ and $v\in C(S)$. we characterize signed graphs on $K_p$, $p\geq2$, on cycle $C_n$ and on $K_{m,n}$ which are $\bullet$-lict signed graphs or $\bullet$-line signed graphs, characterize signed graphs $S$ so that $L_{\bullet_c}(S)$ and $L_\bullet(S)$ are balanced. We also establish the characterization of signed graphs $S$ for which $S\sim L_{\bullet_c}(S)$, $S\sim L_\bullet(S)$, $\eta(S)\sim L_{\bullet_c}(S)$ and $\eta(S)\sim L_\bullet(S)$, here $\eta(S)$ is negation of $S$ and $\sim$ stands for switching equivalence.

The

The line-cut graph (or, in short,

In this paper, we introduce

**Keywords**

**Main Subjects**

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March 2016

Pages 37-48

**Receive Date:**29 September 2013**Revise Date:**01 December 2014**Accept Date:**02 February 2015**Published Online:**01 March 2016