The Line n-sigraph of a Symmetric n-sigraph-V

• Journal title : Kyungpook mathematical journal
• Volume 54, Issue 1,  2014, pp.95-101
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2014.54.1.95
Title & Authors
The Line n-sigraph of a Symmetric n-sigraph-V
Reddy, P. Siva Kota; Nagaraja, K.M.; Geetha, M.C.;

Abstract
An n-tuple ($\small{a_1,a_2,{\ldots},a_n}$) is symmetric, if $\small{a_k}$ = $\small{a_{n-k+1}}$, $\small{1{\leq}k{\leq}n}$. Let $\small{H_n}$ = {$\small{(a_1,a_2,{\ldots},a_n)}$ ; $\small{a_k}$ $\small{{\in}}$ {+,-}, $\small{a_k}$ = $\small{a_{n-k+1}}$, $\small{1{\leq}k{\leq}n}$} be the set of all symmetric n-tuples. A symmetric n-sigraph (symmetric n-marked graph) is an ordered pair $\small{S_n}$ = (G,$\small{{\sigma}}$) ($\small{S_n}$ = (G,$\small{{\mu}}$)), where G = (V,E) is a graph called the underlying graph of $\small{S_n}$ and $\small{{\sigma}}$:E $\small{{\rightarrow}H_n({\mu}:V{\rightarrow}H_n)}$ is a function. The restricted super line graph of index r of a graph G, denoted by $\small{\mathcal{R}\mathcal{L}_r}$(G). The vertices of $\small{\mathcal{R}\mathcal{L}_r}$(G) are the r-subsets of E(G) and two vertices P = $\small{{p_1,p_2,{\ldots},p_r}}$ and Q = $\small{{q_1,q_2,{\ldots},q_r}}$ are adjacent if there exists exactly one pair of edges, say $\small{p_i}$ and $\small{q_j}$, where $\small{1{\leq}i}$, $\small{j{\leq}r}$, that are adjacent edges in G. Analogously, one can define the restricted super line symmetric n-sigraph of index r of a symmetric n-sigraph $\small{S_n}$ = (G,$\small{{\sigma}}$) as a symmetric n-sigraph $\small{\mathcal{R}\mathcal{L}_r}$($\small{S_n}$) = ($\small{\mathcal{R}\mathcal{L}_r(G)}$, $\small{{\sigma}}$'), where $\small{\mathcal{R}\mathcal{L}_r(G)}$ is the underlying graph of $\small{\mathcal{R}\mathcal{L}_r(S_n)}$, where for any edge PQ in $\small{\mathcal{R}\mathcal{L}_r(S_n)}$, $\small{{\sigma}^{\prime}(PQ)}$=$\small{{\sigma}(P){\sigma}(Q)}$. It is shown that for any symmetric n-sigraph $\small{S_n}$, its $\small{\mathcal{R}\mathcal{L}_r(S_n)}$ is i-balanced and we offer a structural characterization of super line symmetric n-sigraphs of index r. Further, we characterize symmetric n-sigraphs $\small{S_n}$ for which $\small{\mathcal{R}\mathcal{L}_r(S_n)}$~$\small{\mathcal{L}_r(S_n)}$ and $\small{\mathcal{R}\mathcal{L}_r(S_n){\sim_=}\mathcal{L}_r(S_n)}$, where ~ and $\small{\sim_=}$ denotes switching equivalence and isomorphism and $\small{\mathcal{R}\mathcal{L}_r(S_n)}$ and $\small{\mathcal{L}_r(S_n)}$ are denotes the restricted super line symmetric n-sigraph of index r and super line symmetric n-sigraph of index r of $\small{S_n}$ respectively.
Keywords
Symmetric n-sigraphs;Symmetric n-marked graphs;Balance;Switching;Restricted super line symmetric n-sigraphs;Super line symmetric n-sigraphs;Complementation;
Language
English
Cited by
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