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

Abstract

A <em>signed graph</em> (or‎, ‎in short‎, <em>sigraph</em>) $S=(S^u,sigma)$ consists of an underlying graph $S^u‎ :‎=G=(V,E)$ and a function $sigma:E(S^u)longrightarrow {+,-}$‎, ‎called the signature of $S$‎. ‎A <em>marking</em> of $S$ is a function $mu:V(S)longrightarrow {+,-}$‎. ‎The <em>canonical marking</em> of a signed graph $S$‎, ‎denoted $mu_sigma$‎, ‎is given as $$mu_sigma(v)‎ :‎= prod_{vwin E(S)}sigma(vw).$$‎ <br />‎The <em>line graph</em> 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 <em>line signed graph</em> 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 $vin V(S)$ is a common vertex of edges $e$ and $e'$‎. <br />‎<br />‎The line-cut graph (or‎, ‎in short‎, <em>lict graph</em>) 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 $c$‎. <br />‎<br />‎In this paper‎, ‎we introduce <em>dot-lict signed graph</em> (or $bullet$<em>-lict signed graph</em>} $L_{bullet_c}(S)$‎, ‎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 $pin V(S)$ is a common vertex of these edges‎, ‎and it has the sign $mu_sigma(v)$‎, ‎if $uin E(S)$ and $vin C(S)$‎. ‎we characterize signed graphs on $K_p$‎, ‎$pgeq2$‎, ‎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 $Ssim L_{bullet_c}(S)$‎, ‎$Ssim L_bullet(S)$‎, ‎$eta(S)sim L_{bullet_c}(S)$ and $eta(S)sim L_bullet(S)$‎, ‎here $eta(S)$ is negation of $S$ and $sim$ stands for switching equivalence‎.