Fundamentals of Fuzzy Graphs

Pal, Madhumangal; Samanta, Sovan; Ghorai, Ganesh

doi:10.1007/978-981-15-8803-7_1

Cited by 1 publication

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“…By adding σ value of a vertex to a particular kind of degree of the same vertex gives the total degree of that vertex of the same kind. (V, σ, μ, η) is edge regular (total edge regular) if edge (total edge) degrees are same for each vertex where as the edge degree 4 of an edge e denoted by ed(e) is the number of edges which share a vertex in common with e. This work mainly follows 6 for the terminologies and preliminaries in graph theory, 7 for fuzzy set concepts and [8][9][10][11] for fuzzy graph theory.…”

Section: Introductionmentioning

confidence: 99%

Line Regular Fuzzy Semigraphs

ARCHANA

KUTTIPULACKAL²

2023

Baghdad Sci.J

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This paper introduce two types of edge degrees (line degree and near line degree) and total edge degrees (total line degree and total near line degree) of an edge in a fuzzy semigraph, where a fuzzy semigraph is defined as (V, σ, μ, η) defined on a semigraph G* in which σ : V → [0, 1], μ : VxV → [0, 1] and η : X → [0, 1] satisfy the conditions that for all the vertices u, v in the vertex set, μ(u, v) ≤ σ(u) ᴧ σ(v) and η(e) = μ(u1, u2) ᴧ μ(u2, u3) ᴧ … ᴧ μ(un-1, un) ≤ σ(u1) ᴧ σ(un), if e = (u1, u2, …, un), n ≥ 2 is an edge in the semigraph G*, in which a semigraph is defined as a pair of sets (V, X) in which the vertex set V is a non - empty set and edge set X is a set of n – tuples for various n ≥ 2, of distinct elements of V with the properties that, any two elements in the edge set X has at most one vertex in common and for any two edges (ɑ1, ɑ2,…, ɑn ) and (b1, b2,…, bm) in the edge set X are equal if, and only if, n = m and either one of the conditions ɑj = bj or ɑj = bn-j+1 occur for j where the value of j lies between 1 and n. In addition to that edge regularities (line regular and near line regular) and total edge regularities (total line regular and total near line regular) of the corresponding edge degrees and total edge degrees are studied, their properties are examined and a few results connecting vertex regularity and edge regularity of a fuzzy semigraph are obtained. 2020 Mathematics Subject Classification: 05C72, 05C07.

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