ON •-LICT SIGNED GRAPHS L•C (S) AND •-LINE SIGNED GRAPHS L• (S)

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ACHARYA MUKTI; JAIN RASHMI; KANSAL SANGITA

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Journal: TRANSACTIONS ON COMBINATORICS Year:2016 | Volume:5 | Issue:1 Page(s): 37-48

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Title

ON •-LICT SIGNED GRAPHS L•C (S) AND •-LINE SIGNED GRAPHS L• (S)

Author(s)

ACHARYA MUKTI | JAIN RASHMI | KANSAL SANGITA | Issue Writer Certificate

Keywords

SIGNED GRAPH

BALANCE

SWITCHING

•-LINE SIGNED GRAPH

•-LICT SIGNED GRAPH

Abstract

A SIGNED GRAPH (or, in short, sigraph) S = (Su, s) consists of an underlying graph Su:= G = (V, E) and a function s: E (Su) ®{+, -}, called the signature of S. A marking of S is a function m: V (S) ®{+, -}. The canonical marking of a SIGNED GRAPH S, denoted ms, is given asms (v):= Õ vw2E(S) s (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 = (Su, s) in the literature, viz., L (S), Lx (S) and L· (S), all of which have L (Su) as their underlying graph, only the rule to assign signs to the edges of L (Su) 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 Lx (S) has the product s (e) s (e′) as its sign and an edge ee′in L· (S) has ms (v) as its sign, where vÎ 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 Lc (G), is the graph with vertex set E (G) È 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 ·-lict SIGNED GRAPH) L·c (S), which has Lc (Su) as its underlying graph. Every edge uv in L·c (S) has the sign ms (p), if u, v Î E (S) and p Î V (S) is a common vertex of these edges, and it has the sign ms (v), if u Î E (S) and v Î C (S). We characterize SIGNED GRAPHs on Kp, p³2, on cycle Cn and on Km, n which are ·-lict SIGNED GRAPHs or ·-line SIGNED GRAPHs, characterize SIGNED GRAPHs S so that L·c (S) and L· (S) are BALANCEd. We also establish the characterization of SIGNED GRAPHs S for which S ~ L·c (S), S ~ L· (S), h (S) ~ L·c (S) and h (S) ~ L· (S), here h (S) is negation of S and ~ stands for SWITCHING equivalence.

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APA: Copy

ACHARYA, MUKTI, JAIN, RASHMI, & KANSAL, SANGITA. (2016). ON •-LICT SIGNED GRAPHS L•_C (S) AND •-LINE SIGNED GRAPHS L• (S). TRANSACTIONS ON COMBINATORICS, 5(1), 37-48. SID. https://sid.ir/paper/689261/en

Vancouver: Copy

ACHARYA MUKTI, JAIN RASHMI, KANSAL SANGITA. ON •-LICT SIGNED GRAPHS L•_C (S) AND •-LINE SIGNED GRAPHS L• (S). TRANSACTIONS ON COMBINATORICS[Internet]. 2016;5(1):37-48. Available from: https://sid.ir/paper/689261/en

IEEE: Copy

MUKTI ACHARYA, RASHMI JAIN, and SANGITA KANSAL, “ON •-LICT SIGNED GRAPHS L•_C (S) AND •-LINE SIGNED GRAPHS L• (S),” TRANSACTIONS ON COMBINATORICS, vol. 5, no. 1, pp. 37–48, 2016, [Online]. Available: https://sid.ir/paper/689261/en

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