LINE REGULAR FUZZY SEMIGRAPHS

: 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(u 1 , u 2 ) ᴧ μ(u 2 , u 3 ) ᴧ … ᴧ μ(u n-1 , u n ) ≤ σ(u 1 ) ᴧ σ(u n ), if e = (u 1 , u 2 , …, u n ), 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 (ɑ 1 , ɑ 2 ,…, ɑ n ) and (b 1 , b 2 ,…, b m ) in the edge set X are equal if, and only if, n = m and either one of the conditions ɑ j = b j or ɑ j = b n-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.


Introduction:
The concept of fuzzy graph was pioneered by A. Rosenfeld 1 . A. Nagoor Gani and K. Radha 2 studied a branch of fuzzy graph theory that deals with the regularity and total regularity of vertices in fuzzy graphs, while T. Nusantara et al. 3 explored the idea of edge degree in fuzzy graphs. In order to improve graph theory and include more scenarios, E. Sampathkumar introduced the concept of semigraph theory 4 . A semigraph 4 G with a vertex set V and an edge set X is defined as the pair of sets (V, X) in which V ≠ φ and X is a set of n -tuples, for various n ≥ 2, of distinct elements of V satisfy the followings, 1. Any two elements in X has atmost one common vertex, 2. Any two edges (ɑ 1 , ɑ 2 ,…, ɑ n ) and (b 1 , b 2 ,…, b m ) are equal if, and only if, n = m and either one of the below conditions occur for j such that 1≤ j ≤ n a. ɑ j = b j b. ɑ j = b n-j+1 A partial edge 4 in a semigraph G is a subedge of an edge in G in which the consecutive vertices in the edge is again consecutive in that subedge. Semigraphs can be of different kinds. A semigraph in which the cardinality of each edge is same is called a uniform semigraph 4 . Integrating the concepts of fuzzy graph theory and semigraph theory K. Radha and P. Renganathan 5 introduced a novel idea called fuzzy semigraph. Let G * = (V, X) be a semigraph. Then (V, σ, μ, η) be the fuzzy semigraph 5 defined on G * in which σ : V → [0, 1], μ : VxV → [0, 1] and η : X → [0, 1] satisfy the conditions that 1. μ(u, v) ≤ σ(u) ᴧ σ(v) for all the vertices u, v in V, 2. η(e) = μ(u 1 , u 2 ) ᴧ μ(u 2 , u 3 ) ᴧ … ᴧ μ(u n-1 , u n ) ≤ σ(u 1 ) ᴧ σ(u n ), if e = (u 1 , u 2 , …, u n ), n ≥ 2 is an edge in G * . Note that ᴧ represents the minimum.
The authors introduced various degrees to the vertices of a fuzzy semigraph and communicated the paper to a journal. That paper discusses, edge degree of a vertex u 1 in a fuzzy semigraph (V, σ, μ, η), denoted by d e (u 1 ) defined to be ∑η(e) where the addition is taken over all of the edges e with the vertex u 1 no matter whether u 1 is an end vertex or a middle vertex. Consecutive adjacent degree of a vertex u 2 , denoted by d ca (u 2 ) defined to be ∑μ(u 2 , u 3 ) where the addition is taken over all u 3 in V which is consecutively adjacent with u 2 in (V, σ, μ, η). 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.

Results:
Two types of edge degrees, namely line degrees and near line degrees in a fuzzy semigraph are defined. The total membership values of the partial edges of cardinality 2 which are consecutively adjacent to the edge E is interpreted as the near line degree of the edge E, denoted by d nl (E).
The addition of membership value of the partial edges of cardinality 2 of E to the near line degree of E defines the total near line degree of the edge E, denoted by d tnl (E).

Example 1:
Consider the edge E = (u 1 , u 2 , u 3 , u 4 ) in the fuzzy semigraph G given in Fig. 1. Note that the edges which are adjacent to the edge E are (u 1 , u 5 , u 6 ), (u 2 , u 7 ) and (u 8 , u 3 , u 9 ) with membership values 0.3, 0.4 and 0.4 respectively and the partial edges of cardinality 2 which are consecutively adjacent to the edge E are (u 1 , u 5 ), (u 2 , u 7 ), (u 3 , u 8  Similarly if the total line degree (total near line degree) of each edge in G is same, then G is called a total line regular (total near line regular) fuzzy semigraph.  290 Observation 1: Consider an edge E = (u 1 , u 2 , …, u m ) of cardinality m in a fuzzy semigraph G = (V, σ, μ, η) with G * = (V, X) as the underlying semigraph. Then Theorem 2: Consider a fuzzy semigraph G = (V, σ, μ, η) in which the function η is constant and the underlying semigraph is uniform. Suppose G is edge regular fuzzy semigraph then G is both line regular and total line regular. Proof: Consider G be a k -edge regular fuzzy semigraph whose underlying semigraph is an runiform. Let η(E) = c for any edge E in G where c is a constant need not be an integer. Let E = (u 1 , u 2 , …, u r ) be an edge in G. Then d l (E) = d l (u 1 , u 2 , …, u r ) = ∑ vєE (d e (v) -η(E)) = r(kc).
Thus, for any edge E in G the line degree is r(kc). Hence G is a line regular fuzzy semigraph.
Since η is constant and G is line regular, the total line degree is r(k -c) + c for any edge in G. Thus G is a total line regular fuzzy semigraph. Hence the result.
The conditions in Theorem 2 is not a sufficient condition for a total edge regular fuzzy semigraph to be line regular or total line regular fuzzy semigraph.

Figure 3. 2 -Uniform fuzzy semigraph
Corollary 1: Consider a fuzzy semigraph G = (V, σ, μ, η) which is both edge regular and total edge regular in which each of its edges are effective and the underlying semigraph is uniform. Then G is line regular as well as total line regular. Proof: Since G is an edge regular and a total edge regular fuzzy semigraph, the function σ is constant. That is G is an effective fuzzy semigraph with a constant σ function. Hence the function η is also constant. Then by Theorem 2 the fuzzy semigraph G is line regular as well as total line regular fuzzy semigraph.
Corollary 2: Consider a fuzzy semigraph G = (V, σ, μ, η) in which G * = (V, X) is the underlying semigraph. In addition G * is an edge regular uniform semigraph. Then G is both line regular and edge regular fuzzy semigraph if, and only if, the function η is constant. Proof: Assume that η is a constant function. Since G * is an edge regular semigraph, G also an edge regular fuzzy semigraph. Then by Theorem 2 the fuzzy semigraph G is line regular. Conversely assume that G is a k 1 -line regular and k 2 -edge regular fuzzy semigraph. Then for any edge E i in G, d l (E i ) = ∑ vєEi (d e (v) -η(E i )) = k 1 ∑ vєEi (k 2 -η(E i )) = k 1 Assume that G * is an r -uniform semigraph. Then rk 2 -r η(E i ) = k 1 η(E i ) = k 2 -(k 1 /r).
Here k 1 , k 2 and r are fixed for any edge in G. Thus, η is a constant function in G.
Theorem 3: Consider a fuzzy semigraph G = (V, σ, μ, η). Then the following condition are analogous, 1. G is a line regular fuzzy semigraph 2. G is a total line regular fuzzy semigraph if, and only if, η is a constant function. Proof: Assume that η is a constant function. Let η(E) = c for an edge E in G where c is a constant need not be an integer. Assume G is line regular. Thus, for any edges E i and E j in G, d l (E i ) = d l (E j ). Consequently d l (E i ) + η(E i ) = d l (E j ) + η(E j ). That is d tl (E i ) = d tl (E j ) for any edges E i and E j. Hence G is a total line regular fuzzy semigraph. Next assume G is total line regular. Hence for any edges E i and E j , d tl (E i ) = d tl (E j ), which implies d l (E i ) = d l (E j ). That is G is a line regular fuzzy semigraph.
Conversely assume that the given statements are equivalent. Suppose that η is a non-constant function. That is one can find atleast a pair of edges E i and E j in G such that η(E i ) ≠ η(E j ). Assume G is line regular. So that the edges E i and E j satisfies d l (E i ) = d l (E j ). But d l (E i ) + η(E i ) ≠ d l (E j ) + η(E j ), which gives G is not a total line regular fuzzy semigraph. Reached a contradiction. Now assume G is total line regular. Here the edges E i and E j satisfies d tl (E i ) = d tl (E j ), which implies d l (E i ) + η(E i ) = d l (E j ) + η(E j ). This hold only if d l (E i ) ≠ d l (E j ), which shows G is not a line regular fuzzy semigraph, again reached a contradiction. Thus, η must be a constant function.
Theorem 4: Consider a fuzzy semigraph G = (V, σ, μ, η). Suppose G is a line regular and a total line regular fuzzy semigraph then the function η is constant. Moreover, the constant value is the difference between regularity and total regularity of G. Proof: Consider G be a k 1 -line regular and k 2total line regular fuzzy semigraph. Then for any edges E 1 and E 2 in G, d tl (E 1 ) = d l (E 1 ) + η(E 1 ) = k 2 = d tl (E 2 ) = d l (E 2 ) + η(E 2 ).
Also note that d l (E 1 ) = k 1 = d l (E 2 ). Thus η(E 1 ) = η(E 2 ) = k 2k 1 , where k 1 and k 2 are fixed. Thus η is a constant function and the constant value is the difference between the line regularity and total line regularity of G.
The converse is not true. For, consider the fuzzy semigraph G in the Fig. 4 where the function η, which is the membership value of the edges has a constant value 0.2. Here G is neither line regular nor total line regular.

Figure 4. A fuzzy semigraph G
The following corollary is obvious in view of Theorem 4.

Corollary 3:
Consider a fuzzy semigraph G = (V, σ, μ, η) in which G is both line regular and total line regular fuzzy semigraph. Then the underlying semigraph of G is edge regular if, and only if, the fuzzy semigraph G itself is edge regular.
With the help of 6. and 7. in the Observation 1, the below result holds.

Conclusion:
In this research, the line degree, near line degree, total line degree, and total near line degree of a fuzzy semigraph are introduced. Additionally, relevant regularities and total regularities are investigated. Despite the fact that the regularities cannot be generally contrasted, an analysis is carried out by restricting the properties of the fuzzy semigraph.