# ON $\bullet$-LICT signed graohs $L_{\bullet_c}(S)$ and $\bullet$-LINE signed graohs $L_\bullet(S)$

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.

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