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STRENGTH of CERTAIN FUZZY GRAPHS K.P. Chithra1 §, Raji International Journal of Pure and Applied Mathematics Volume 106 No. 3 2016, 883-892 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu AP doi: 10.12732/ijpam.v106i3.14 ijpam.eu STRENGTH OF CERTAIN FUZZY GRAPHS K.P. Chithra1 §, Raji Pilakkat2 1Department of Mathematics University of Calicut INDIA Abstract: In this paper we find the strength of a strong fuzzy wheel graph, strong fuzzy complete bipartite graph, strong fuzzy butterfly graph, strong fuzzy bull graph and also de- termined the strength of a properly linked fuzzy graph with complete fuzzy graphs as its parts. AMS Subject Classification: 05C72 Key Words: strength of fuzzy graphs, fuzzy wheel graph, fuzzy hub, fuzzy butterfly graph, n-linked fuzzy graphs, fuzzy bull graph 1. Introduction In 1965, Lofti A. Zadeh [7] introduced the notion of a fuzzy subset. Azriel Rosenfeld [5] defined fuzzy graph based on the definitions of fuzzy sets and fuzzy relations and developed the theory of fuzzy graphs in 1975. John N. Mordeson and Premchand S. Nair [3] introduced different types of operations on fuzzy graphs. Sheeba M.B.[6] introduced the concept of strength of fuzzy graphs. She determined the strength of fuzzy graphs in two different ways. One by introducing weight matrix of a fuzzy graph and the other by introducing the concept of extra strong path between its vertices. In this paper we use the concept of extra strong path to find the strength of various fuzzy graphs. Throughout this paper only undirected fuzzy graphs are considered. Received: October 19, 2015 c 2016 Academic Publications, Ltd. Published: February 27, 2016 url: www.acadpubl.eu §Correspondence author 884 K.P. Chithra, R. Pilakkat 1.1. Preliminaries A fuzzy graph G = (V, µ, σ) [3] is a nonempty set V together with a pair of functions µ : V −→ [0, 1] and σ : V × V −→ [0, 1] such that for all u, v ∈ V , σ(u, v) = σ(uv) ≤ µ(u) ∧ µ(v). We call µ the fuzzy vertex set of G and σ the fuzzy edge set of G. Here after we denote the fuzzy graph G(µ, σ) simply by G and the underlying crisp graph of G by G∗(V, E) with V as vertex set and E = {(u, v) ∈ V × V : σ(u, v) > 0} as the edge set or simply by G∗. If for (u, v) ∈ E we say that u and v are adjacent in G∗. In that case we also say that u and v are adjacent in G. A fuzzy graph G is complete [3] if σ(uv) = µ(u)∧µ(v) for all u, v ∈ V . A fuzzy graph G is a strong fuzzy graph [3] if σ(uv) = µ(u)∧µ(v), ∀uv ∈ E. The strength of a strong fuzzy complete graph is one [6]. Let G1(V1, µ1, σ1) and G2(V2, µ2, σ2) be two fuzzy graphs with the underlying crisp graphs G1(V1,X1) and G2(V2,X2) respectively. If V1 ∩ V2 = φ then their join [3] is the fuzzy graph G = G1 ∨ G2(V1 ∪ V2, µ1 ∨ µ2, σ1 ∨ σ2) with ∗ ′ ′ the underlying crisp graph G = (V1 ∪ V2,X1 ∪ X2 ∪ X ) where X is the set of all edges joining the vertices of V1 and V2 and µ1(u) if u ∈ V1 \ V2, (µ1 ∨ µ2)(u) = µ2(u) if u ∈ V2 \ V1; σ1(uv) if uv ∈ X1 \ X2, (σ1 ∨ σ2)(uv) = σ2(uv) if uv ∈ X2 \ X1, µ1(u) ∧ µ2(v) if u ∈ X1 and v ∈ X2. A strong fuzzy complete bipartite graph is a strong fuzzy graph with its underlying crisp graph is a complete bipartite graph [4]. A path P of length n − 1 in a fuzzy graph G [3] is a sequence of distinct vertices v1, v2, v3, . , vn, such that σ(vi, vi+1) > 0, i = 1, 2, 3, . , n − 1. We call P a fuzzy cycle if v1 = vn and n ≥ 3 and there does not exist a unique edge such that σ(xy) = ∧{σ(uv)/(uv) > 0}. The strength of a path is defined as the weight of the weakest edge of the path [3]. A path P is said to connect the vertices u and v of G strongly if its strength is maximum among all paths between u and v. Such paths are called strong paths [7]. Any strong path between two distinct vertices u and v in G with minimum length is called an extra strong path between them [6]. There may exists more than one extra strong paths between two vertices in a fuzzy graph G. But, by the definition of an extra strong path each such path between two vertices has the same length. The maximum length of extra strong paths between every pair of distinct vertices in G is called the strength ∗ of the graph G [6]. For a fuzzy graph G, if G is the path P = v1v2 . vn on n vertices then the strength of the graph G is its length (n − 1) [6]. STRENGTH OF CERTAIN FUZZY GRAPHS 885 Here after for a fuzzy graph G we use S (G) to denote its strength. The following theorems determine the strength of a fuzzy cycle. Theorem 1. [6] In a fuzzy cycle G of length n, suppose there are l weakest n+1 edges where l ≤ [ 2 ]. If these weakest edges altogether form a subpath then S (G) is n − l. Theorem 2. [6] Let G be a fuzzy cycle with crisp graph G∗ a cycle of n+1 length n, having l weakest edges which altogether form a subpath. If l > [ 2 ], S n then (G) is [ 2 ]. Theorem 3. [6] Let G be a fuzzy cycle with crisp graph G∗ a cycle of length n, having l weakest edges which do not altogether form a subpath. If n n n S l > [ 2 ] − 1 then the strength of the graph is [ 2 ] and if l = [ 2 ] − 1 then (G) n+1 is [ 2 ]. n Theorem 4. [6] In a fuzzy cycle of length n suppose there are l < [ 2 ] − 1 weakest edges which do not altogether form a subpath. Let s denote the maximum length of a subpath which does not contain any weakest edge. If n n n s ≤ [ 2 ] then the strength of the graph is [ 2 ] and if s > [ 2 ] then the strength of the graph is s. 2. Main Results Lemma 5. Let G be a strong fuzzy graph. If u and v are two adjacent vertices of G then the length of the extra strong path joining u and v is one. Proof. Suppose that u and v are adjacent in G. Since G is a strong fuzzy graph, the edge uv, which is a path u − v joining u and v, has strength µ(u) ∧ µ(v). All the other paths joining u and v have strength less than or equal to µ(u) ∧ µ(v). Hence the edge uv is the unique extra strong path joining u and v. Hence the result. Remark 1. Let G be a fuzzy graph. Suppose u and v are two adjacent vertices of G. Then the edge uv is the only extra strong path joining u and v. So for the computation of strength of a fuzzy graph we need only to consider its distinct non-adjacent vertices. We introduce the following definitions to get a better understanding of the proof of the theorems that we are going to prove. Definition 6. A fuzzy graph with K1 [2] as its underlying graph is called a fuzzy vertex and it is also denoted by K1. 886 K.P. Chithra, R. Pilakkat Definition 7. A fuzzy wheel graph Wn is the join of the fuzzy cycle Cn−1 and the fuzzy vertex K1. Definition 8. A vertex h of the wheel graph Wn is said to be a fuzzy hub if it is adjacent to all the other vertices of Wn. Definition 9. A strong fuzzy wheel graph is a fuzzy wheel graph which is also a strong fuzzy graph. Theorem 10. For n ≥ 4, let Wn = Cn−1 ∨ K1 be a strong fuzzy wheel graph with the fuzzy hub h and u1u2 . un−1u1 the fuzzy cycle Cn−1. If µ(h) < µ(ui), ∀i = 1, 2, . , n − 1 then the strength of Wn is the strength of Cn−1. Proof. Since h is adjacent to all the ui s, i = 1, 2, . , n − 1, length of all h − ui paths are one by Lemma 5. Choose two distinct non-adjacent vertices ui and uj of Cn−1. Since µ(h) < µ(ui), i = 1, 2, . , n − 1, all paths joining ui and uj, through h have strength µ(h), which is less than the strength of any path joining ui and uj in Cn−1. Therefore the length of extra strong paths joining ui and uj in Wn and those in Cn−1 are one and the same. Hence the result. Theorem 11. Let Wn be as in Theorem 10. If µ(h) ≥ µ(ui), ∀i = 1, 2, . , n − 1 then the strength of Wn is one when n = 4 and two when n > 4. Proof. When n = 4, Wn is a complete fuzzy graph. Therefore the strength of Wn is one [6]. Now suppose that n > 4. Let u and v be any two distinct non-adjacent vertices of Wn. Therefore both belong to V (Cn−1) and the length of any extra strong path joining u and v is greater than or equal to 2. Without loss of generality assume that µ(u) ≤ µ(v).
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