Author/Authors :
Acharya، Mukti نويسنده Department of Applied Mathematics,DELHI TECHNOLOGICAL UNIVERSITY,DELHI,India , , Jain، Rashmi نويسنده Department of Applied Mathematics,DELHI TECHNOLOGICAL UNIVERSITY,DELHI,India , , Kansal، Sangita نويسنده DELHI TECHNOLOGICAL UNIVERSITY,,Department of Applied Mathematics,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 $$mu_sigma(v) := prod_{vwin E(S)}sigma(vw).$$ 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#039 $ in $L(S)$ is negative whenever both the adjacent edges $e$ and $e#039 $ in S are negative, an edge $ee#039 $ in $L_times(S)$ has the product $sigma(e)sigma(e#039 )$ as its sign and an edge $ee#039 $ in $L_bullet(S)$ has $mu_sigma(v)$ as its sign, where $vin V(S)$ is a common vertex of edges $e$ and $e#039 $. The linecut 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 cutvertices 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 cutvertex $c$ of $G$ such that $e$ is incident with $c$. In this paper, we introduce dotlict 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 $pin V(S)$ 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.
Keywords :
Balance , switching , •-line signed graph , •-lict signed graph. , Signed graph
