OPERATIONS ON S -GRAPHS

: In our earlier paper (see [4]), we have introduced the notion of semiring-valued graphs and (see [3]) the notion of regularity on S -graphs. In this paper, we introduce and study some operations such as union and sum of two S -graphs.


Introduction
Eventhough the concept of semiring was first introduced by H.S.Vandiver [5] in 1934, the developments of the theory in semirings and ordered semirings have been taking place since 1950.Jonathan S.Golan [2] has introduced the notion of S-graph where he considers a function g : V × V → S such that g(v 1 , v 2 ) = 0.But nothing more has been dealt.This motivated us to study graphs whose vertices and edges are assigned values from the semiring S [4].Golan considers the S-graph by assigning values to edges only.However we assign values to every vertex of the graph and the weights of an edge is assigned in relation to the weights of the vertices incident with the edges.Since every semiring possesses a canonical pre-order, for any edge e = (v i , v j ), we can assign the weight of e as the minimum weights of v i and v j .Such a graph we called a S-graph.In our paper [3], we studied the notion of regularity on S-graphs.In this paper, we introduce and study the notion of operations such as union and sum of two S-graphs.

Preliminaries
In this section, we recall the basic definitions on operations on crisp graphs, such as union and sum of two graphs.For graph theoretical concepts, we refer [1].
Definition 2.3.A semiring (S, +, •) is an algebraic system with a nonempty set S together with two binary operations + and • such that 1. (S, +, 0) is a monoid.

For all
Definition 2.4.Let S be a semiring.is said to be a Canonical Pre-order if for a, b ∈ S , a b if and only if there exists c ∈ S such that a + c = b.Definition 2.5.[4] Let G = (V, E ⊂ V × V ) be the underlying graph with both V, E = φ.For any semiring (S, +, •), a Semiring-valued graph (or an Svalued graph) G S is defined to be the graph G S = (V, E, σ, ψ) where σ : V → S and ψ : E → S is defined to be ψ(x, y) = min {σ(x), σ(y)} if σ(x) σ(y) or σ(y) σ(x) 0 otherwise for every unordered pair (x, y) of E ⊂ V × V. We call σ, a S-vertex set and ψ, a S-edge set of S-valued graph G S .Henceforth we call a S-valued graph simply as a S-graph.
Definition 2.6.[4] If σ(x) = a, ∀ x ∈ V and some a ∈ S then the corresponding S-graph G S is called a vertex regular S-graph (or simply vertex regular).Definition 2.7.[4] An S-graph G S is said to be an edge regular S-graph (or simply edge regular) if ψ(x, y) = a for every (x, y) ∈ E and some a ∈ S. Definition 2.8.[4] An S-graph G s is said to be S-regular if it is both vertex regular and edge regular .Definition 2.9.[3] Let G S be an S-graph corresponding to an underlying graph G. G S is said to be (a, k)−regular S-graph if the following conditions are true.

Operations on S-Graph
In this section, we define the notion of union and sum of two S-graphs and prove some simple results.Let G 1 and G 2 be two crisp graphs given by : . Here Since the existance of an S-graph corresponding to its underlying graph is not unique, we define S and the equality holds onlyif σ = σ 3 .Lemma 3.4.Union of two vertex regular S-graphs is a vertex regular Sgraph if and only if their corresponding S-vertex sets are constant and assigns the same value in S.
2 ) be two vertex regular S-graphs, corresponding to the underlying graphs 1 and G S 2 are vertex regular, σ 1 (v) = a for every v ∈ V 1 , and some a ∈ S and σ 2 (v) = b for every v ∈ V 2 , and some b ∈ S,respectively.Hence σ 1 and σ 2 are constant.Let G S 1 ∪ G S 2 = (V, E, σ, ψ) be the union of given S-graphs and let it be vertex regular.Therefore σ(v) = c for every v ∈ V , and some c ∈ S.
Claim.σ 1 and σ 2 assigns the same value in S.
Hence σ 1 and σ 2 are constant and assigns the same value in S. Conversely, Let σ 1 and σ 2 be constant and assigns the same value in S. Let a ∈ S. and σ Let v ∈ V be arbitrary.Then Remark 3.8.Clearly sum of two S-graphs is also an S-graph.Lemma 3.9.Sum of two vertex regular S-graphs is an vertex regular S-graph iff their corresponding S-vertex sets are constant and assigns the same value in S.
2 ) be two vertex regular S-graphs, corresponding to the underlying graphs and some a ∈ S then σ 1 is constant and σ 2 (v) = b for every v ∈ V 2 , and some b ∈ S then σ 2 is constant.
Let G S 1 +G S 2 = (V, E, σ, ψ) be the sum of given S-graphs and let it be vertex regular S-graph.
Therefore σ(v) = c for every v ∈ V, and some c ∈ S.
Claim.σ 1 and σ 2 assigns the same value in S.
Hence σ 1 and σ 2 assigns the same value in S. Conversely, assume that σ 1 and σ 2 are constant and assign the same value in S. Let a ∈ S and σ Let v ∈ V be arbitrary.Then That is σ(v) = a, ∀v ∈ V and a ∈ S. Therefore G S 1 + G S 2 is a vertex regular S-graph.Corollary 3.10.Sum of two vertex regular S-graphs is S-regular if and only if their corresponding S-vertex sets are constant and assigns the same value in S. Let G S 1 and G S 2 be two edge regular but not vertex regular S-graphs given by : Here Here ψ(v i , v j ) = a, for all (v i , v j ) ∈ E. Therefore G S 1 + G S 2 is not an edge regular S-graph.
Theorem 3.12.Sum of two edge regular S-graphs is an edge regular S-graph only if their corresponding S-vertex sets are constants and assigns the same value in S.
) be two edge regular S-graphs, corresponding to the underlying graphs Since G S 1 and G S 2 are edge regular.ψ 1 (v i , v j ) = a for every (v i , v j ) ∈ E 1 , and some a ∈ S. ψ 2 (v i , v j ) = b for every (v i , v j ) ∈ E 2 , and some b ∈ S.

Claim. G S
1 + G S 2 is an edge regular S-graph only if their corresponding S-vertex sets assigns the same value in S.
Case 1. Suppose the S-vertex sets of G S 1 and G S 2 are constants and assigns the same value.
By corollary 3.10 G S 1 + G S 2 is a S-regular graph.Therefore it is an edge regular S-graph.

Case 2. Suppose G S
1 and G S 2 are edge regular and corresponding S-vertex sets are constant and σ 1 (v i ) = a f orsome a ∈ S and f or every v i ∈ V 1 ; and σ 2 (v j ) = b f orsome b ∈ S and f or every v j ∈ V 2 such that a = b.Then for(v i , v j ) ∈ E, Since the edges in E 1 and E 2 assumes different values, we have ψ(v i , v j ) is not a constant for every (v i , v j ) ∈ E.
Therefore G S 1 + G S 2 is not an edge regular S-graph.Case 3. Suppose G S 1 and G S 2 are edge regular and corresponding S-vertex sets are not constant.Then there exists some v k ∈ V 1 such that σ 1 (v k ) = c for some c ∈ S, and c = a.
Similarly there exists some v l ∈ V 2 such that σ 2 (v l ) = d for some d ∈ S, and d = b.Clearly (v l , v k ) or (v k , v l ) ∈ E. Then

Example 3 . 3 .
Let (S = {0, a, b} , +, •) be a semiring with the following Cayley Tables: + 0 a b 0 0 a b a a o b b b b b • 0 a b 0 0 0 0 a 0 0 0 b 0 0 b Clearly is a canonical pre-order in S, where 0 0, 0 a, 0 b, a a, a 0, a b, b b.

Remark 3 .
11.If G S 1 and G S 2 are edge regular S-graphs then their sum need not be, in general, an edge regular S-graph.Proof.Let (S = {0, a, b} , +, •) be a semiring with the following Cayley TablesClearly is a canonical pre-order in S, where 0 0, 0 a, 0 b, a a, a 0, a b, b b.