Irregular Interval Valued Fuzzy Graphs

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1 nnals of Pure and pplied Mathematics Vol 3, No, 03, ISSN: X (P), (online) Published on 0 May 03 wwwresearchmathsciorg nnals of Irregular Interval Valued Fuzzy Graphs Madhumangal Pal and Hossein Rashmanlou Department of pplied Mathematics with Oceanology and Computer Programming Vidyasagar University, Midnapore-70, India mmpalvu@gmailcom Department of Mathematics, University of Mazandaran, abolsar, Iran hrashmanlou@yahoocom Received 6 December 0; accepted pril 03 bstract In this paper, we define irregular interval-valued fuzzy graphs and their various classifications Size of regular interval-valued fuzzy graphs is derived The relation between highly and neighbourly irregular interval-valued fuzzy graphs are established Some basic theorems related to the stated graphs have also been presented Keywords: Interval-valued fuzzy graphs, irregular interval-valued fuzzy graphs, totally irregular interval-valued fuzzy graph MS Mathematics Subject Classification (00): 05C7 Introduction Presently, science and technology is featured with complex processes and phenomena for which complete information is not always available For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty large number of these models are based on an extension of the ordinary set theory, namely, fuzzy sets In 965, Zadeh [3] introduced the notion of fuzzy subset of a set as a method of presenting uncertainty The fuzzy systems have been used with success in last years, in problems that involve the approximate reasoning It has become a vast research area in different disciplines including medical and life science, management sciences, social sciences, engineering, statistics, graph theory, artificial intelligence, signal processing, multi agent systems, pattern recognition, robotics, computer networks, expert systems, decision making, etc In 975, Rosenfeld [9] introduced the concept of fuzzy graphs In 975, Zadeh [30] introduced the notion of interval-valued fuzzy sets as an extension of fuzzy sets [3] in which the values of the membership degrees are intervals of numbers instead of real numbers between 0 and Interval-valued fuzzy sets provide a more adequate description of uncertainty than traditional fuzzy sets It is therefore important to use interval-valued fuzzy sets in applications, such as fuzzy control One of the computationally most intensive parts of fuzzy control is defuzzification [0] Since interval-valued fuzzy sets 56

2 Irregular Interval Valued Fuzzy Graphs are widely studied and used, we describe briefly the work of Gorzalczany on approximate reasoning [8-9], Roy and iswas on medical diagnosis [0], Turksen on multivalued logic [5] and Mendel on intelligent control [0] In 0, kram [] introduced the concept of interval-valued fuzzy graphs and defined different operations on it Interval- valued fuzzy graph theory is now growing and expanding its applications The theoretical development in this area is discussed here Review of literature fter Rosenfeld [9], fuzzy graph theory is increased with a large number of branches Mathew and Sunitha [] described the types of arcs in a fuzzy graph Nagoorgani and Malarvizhi [4] established the isomorphism properties of strong fuzzy graphs Nagoorgani and Vadivel [5] established relations between the parameters of independent domination and irredundance in fuzzy graphs Nair and Cheng [7] defined cliques and fuzzy cliques in fuzzy graphs Nair [8] established the definition of perfect and precisely perfect fuzzy graphs kram [] defined different operations on bipolar fuzzy graphs Strong bipolar fuzzy graphs were also introduced here He also introduced regular bipolar fuzzy graphs [4] Samanta and Pal introduced fuzzy tolerance graphs [], fuzzy threshold graphs [3], fuzzy competition graphs [] and bipolar fuzzy hypergraph [4] kram and Davvaz discussed the properties of strong intuitionistic fuzzy graphs [3] Talebi and Rashmanlou [6] studied isomorphism on interval-valued fuzzy graph Likewise, they defined isomorphism on vague graphs [7] very few algorithms have also been designed to solve problems on fuzzy graphs [3,33,34] Preliminaries fuzzy set on a set X is characterized by a mapping m: X [0, ], called the membership function fuzzy set is denoted as (X, m) fuzzy graph [9] ξ ( V, σ, µ ) is a non-empty set V together with a pair of functions σ :V [0, ] and µ :V V [0, ] such that for all u, v V, µ ( u, σ ( u) σ ( (here x y denotes the minimum of x and y) Partial fuzzy subgraph ξ ( V, τ, of ξ is such that τ ( σ ( for all v V and µ ( u, V ( u, for all u, v V Fuzzy subgraph [] ξ ( P, σ, µ ) of ξ is such that P V, σ ( u) σ ( u) for all u P, µ ( u, µ ( u, for all u, v P fuzzy graph is complete [3] if µ ( u, σ ( u) σ ( for all u, The degree of vertex u is d ( u) µ ( u, The minimum degree of ξ is δ (ξ ) { d( u) u V} ( u, ξ The maximum degree of ξ is ) { d( u) u V} (ξ The total degree [3] of a vertex u V is td( u) d( u) σ ( u) fuzzy graph ξ ( V, σ, µ ) is said to be regular [3] if d ( k, a positive real number, for all v V If each vertex of ξ has same total degree k, then ξ is said to be a totally regular fuzzy graph fuzzy graph is said to be irregular [6], if there is a vertex which is adjacent to vertices with distinct degrees fuzzy graph is said to be neighbourly irregular [6], if every two adjacent vertices of the graph have different degrees 57

3 Madhumangal Pal and Hossein Rashmanlou fuzzy graph is said to be totally irregular, if there is a vertex which is adjacent to vertices with distinct total degrees If every two adjacent vertices have distinct total degrees of a fuzzy graph then it is called neighbourly total irregular [6] fuzzy graph is called highly irregular [6] if every vertex of G is adjacent to vertices with distinct degrees The complement [] of fuzzy graph ξ ( V, σ, µ ) is the fuzzy graph ξ ( V, σ, µ ) where σ ( u) σ ( u) for all u V and 0 µ ( u, σ ( u) σ (, 58 if µ ( u, > 0, otherwise Let X be a nonempty set n interval-valued fuzzy set in V is defined by x,[ µ, ]) x V, where µ and µ are fuzzy subsets of { : } ( µ ( x) ( x ) V such that µ µ for all x V ( x) ( x) (x) (x) For any two interval-valued fuzzy sets µ, µ ] and [ ( x) ( x ) µ, µ ] in V we have: [ ( x) ( x ) U {( x, max( µ ( x), µ ( x)), max( µ ( x), ( x))) : x V}, µ x, min( µ ( x), µ ( x)), min( µ ( x), µ I {( ( x))) : x V } If G ( V, E) is a graph, then by an interval-valued fuzzy relation on a set E we mean an interval - valued fuzzy set such that µ ( xy) min ( µ ( x), µ ( y) ), ( µ ( x), ( )) µ ( xy) min µ y for all xy E y an interval - valued fuzzy graph of a graph G ( V, E) we mean a pair G (, ), where µ, µ ] is an interval-valued fuzzy set on V and µ, µ ] is an [ [ interval-valued fuzzy relation on E The graph G is called complete interval-valued fuzzy graph if ( xy) min µ ( x), µ ( y) µ ( xy) min µ ( x), µ ( y) for all µ ( ) and ( ) x, y V * n interval-valued fuzzy graph G (, ) of a given graph G ( V, E) is called an interval-valued strong fuzzy graph if ( xy) min µ ( x), µ ( y) µ ( xy) min µ ( x), µ ( y) for all µ ( ) and ( ) xy E The complement of a strong interval-valued fuzzy graph G is G (, ) where [ µ ( x), µ ( x)] is an interval-valued fuzzy set on V and µ, µ ] is an interval-valued fuzzy set on E V V such that () V V, () µ ( x) µ ( x) and µ ( x) µ ( x) for all x V, [

4 Irregular Interval Valued Fuzzy Graphs µ (3) (xy) µ ) (4) (xy 0 µ 0 µ ( x) µ ( x) µ ( y) ( y) If If If If µ ( xy) > 0, µ ( xy) 0 µ ( xy) > 0, µ ( xy) 0 Definition [] Let G (, ) be an interval-valued fuzzy graph where µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty [ finite set V and [ E V V respectively The graph G is called complete interval- µ ( xy) min µ ( x), µ ( y) and valued fuzzy graph if ( ) ( xy) min ( µ ( x), µ ( y) ) µ for all x, y V Definition Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval - valued fuzzy sets on a non- empty finite set V and [ E V V respectively The total degree of a vertex u V is denoted by td(u) and defined as td( u) [ td ( u), td ( u)] where td ( u) µ ( u µ ( u), td ( u) µ ( u µ ( u) If the total degrees of all vertices of an interval-valued fuzzy graph are equal, then the graph is said to be totally regular interval-valued fuzzy graph 3 Some definitions related to interval-valued fuzzy graphs The degree of a vertex of an interval-valued fuzzy graph is defined below Definition 3 Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E V V respectively The positive degree of a vertex u G is d ( u) µ ( u Similarly, negative degree of a vertex u G is d u) ( u ( µ The degree of a vertex u is d( u) [ d ( u), d ( u)] If d ( u) k, d ( u) k for all u V, k, k are two real numbers, then the graph is called k, ]-regular interval valued fuzzy graph [ k 59

5 Madhumangal Pal and Hossein Rashmanlou Example We consider an interval-valued fuzzy graph show here We have d ( x) 0 0 0, d ( x) So, d ( x) (0, 07) Similarly, d ( y) (03, 07) and d ( z) (03, 08) x y [0, 03] [0, 04] [03, 05] [0, 04] [0, 04] [04, 05] z Figure : n example of interval-valued fuzzy graph The order and size of an interval-valued fuzzy graph are important terms They are defined below Definition 4 Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and [ µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and E V V respectively The order of G is denoted by O(G) and is defined by O( G) [ O ( G), O ( G)] where O ( G) µ ( u) and O ( G) µ ( u) u V u V [ µ, µ Definition 5 Let G (, ) be an interval-valued fuzzy graph where ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E respectively The size of G is defined by S( G) [ S ( G), S ( G)] where S ( G) µ ( u and S ( G) µ ( u u v u v Example For the interval-valued fuzzy graph of Figure, O ( G) (09,4) and S ( G) (04,) 60

6 Irregular Interval Valued Fuzzy Graphs Definition 6 Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E V V respectively The underlying crisp graph of G is the crisp graph G ( V, E ) where V { v µ ( > 0 and ( > 0} and E µ { ( u, µ ( u > 0 and µ ( u > 0} Definition 7 n interval-valued fuzzy graph is said to be connected if it s underlying crisp graph is connected Theorem Let G be a regular interval-valued fuzzy graph where induced crisp graph G is an even cycle Then G is regular interval-valued fuzzy graph if and only if either µ or µ is constant functions or alternate edges have same positive membership values and negative membership values Proof Let G (, ) be a regular interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E V V respectively and underlying crisp graph G of G be an even cycle If either µ or µ is constant function or alternate edges have same positive and negative membership values, then G is a regular interval-valued fuzzy graph Conversely, suppose G is a [ k, k ]-regular interval-valued fuzzy graph Let e, e, e n be the edges of G in order s c if i is odd, µ ( e ) i k c if i is even c if i is odd, µ ( e ) i k c if i is even If c k c, then µ is constant If c, k c then alternate edges have same positive and negative membership values Similarly, for µ Hence the result Theorem The size of a ( k, k) -regular interval-valued fuzzy graph is ( Pk, Pk ) where P V Proof Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E V V respectively The size of G is S ( G) µ ( u, µ ( u u v u v 6

7 Now, Madhumangal Pal and Hossein Rashmanlou d ( µ µ S( G) [ ( u, ( u] S ( G) d(, i e S( G) [ k, k ( G) [ P k, P k Thus, ] This gives S ] Hence the result Theorem 3 If G is [ k, k ] -totally regular interval-valued fuzzy graph, then S ( G) O( G) [ P k, Pk ] where P V Proof Let G (, ) be an interval-valued fuzzy graph where u v and µ ( uv ), µ ( u be two interval-valued fuzzy sets on a non-empty finite set V u v V V respectively Since G is a [ k, k ] -totally regular interval-valued fuzzy graph So k td ( d ( µ ( and k td ( d ( µ ( for all v V Therefore ( µ ( v and d k d ) Pk S ( G) and Pk S ( G) So Pk Pk ( S ( G) S ( G)) O ( G) O ( G) Hence S ( G) O( G) [ Pk, Pk ] k ( µ ( 4 Irregular interval-valued fuzzy graphs Irregular interval-valued fuzzy graphs are important as regular interval-valued fuzzy graphs We now define it Definition 8 Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E V V respectively G is said to be irregular interval-valued fuzzy graph if there exists a vertex which is adjacent to a vertex with distinct degrees Example 3 Let G (, ) be an interval - valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and E V V [ respectively, where { v, v, v v } V 3, d ( v ) [08,], d ( v ) [08,], d ( v 3 ) [,4], d ( v 4 ) [04, 0] Here d( V ) d( V3 ) So this graph is an example of irregular interval-valued fuzzy graph, show in Figure Neighbourly irregular interval-valued fuzzy graph is a special case of irregular interval-valued fuzzy graph 4 6

8 Irregular Interval Valued Fuzzy Graphs Definition 9 Let G be a connected interval-valued fuzzy graph Then G is called neighbourly irregular interval-valued fuzzy graph if for every two adjacent vertices of G have distinct degrees v v4 [04, 05] [05, 06] [04, 05] [04, 05] [04, 04] [05, 07] [04, 07] [04, 05] v v 3 Figure : n example of irregular interval-valued fuzzy graph Definition 0 Let G (, ) be an interval-valued fuzzy graph where µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty [ [ finite set V and E V V respectively G is said to be totally irregular interval-valued fuzzy graph if there exists a vertex which is adjacent to a vertex with distinct total degrees Definition Let G be a connected interval-valued fuzzy graph Then G is called highly irregular interval-valued fuzzy graph if every vertex of G is adjacent to vertices with distinct degrees Example 4 Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E V V V 3, 4 v [04, 05 ], v [05, 06 ], v [04, 3 05], v [04, 04], ( µ ) v v, µ ( v ) 0 4 v, µ ( v 0, v3 ) µ ( v ) 3 04, µ ( v v ) 4 0 3, µ v ) 0 4, µ ( v v ) 4 0, respectively, where { v, v, v v } v ( v v4 ) µ 04 3 ( 3 v4 So we have d ( v ) [03, 04 ], d ( v ) [08, ], d ( v 3 ) [05, 08 ], d ( v 4 ) [05, 08]) Here the interval-valued fuzzy graph is highly irregular but not neighbourly irregular as d v ) d( ) ( 3 v4 63

9 Madhumangal Pal and Hossein Rashmanlou Theorem 4 Let G be an interval-valued fuzzy graph Then G is highly irregular intervalvalued fuzzy graph and neighbourly irregular interval-valued fuzzy graph if and only if the degrees of all vertices of G are distinct Proof Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and 64 µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and V V [ respectively Let V { v v,, } We assume that G is highly irregular and, v n neighbourly irregular interval-valued fuzzy graphs Let the adjacent vertices of u be u, u3,, u n with degrees [ k, k ],[ k3, k3 ],,[ k n, k n ] respectively s G is highly and neighbourly irregular, d( u) d( u) d( u3) d( un) So it is obvious that all vertices are of distinct degrees Conversely, assume that the degrees of all vertices of G are distinct This means that every two adjacent vertices have distinct degrees and to every vertex the adjacent vertices have distinct degrees Hence, G is neighbourly irregular and highly irregular intervalvalued fuzzy graphs Theorem 5 Let G be an interval-valued fuzzy graph If G is neighbourly irregular and µ, µ are constant functions, then G is a neighbourly total irregular interval-valued fuzzy graph Proof Let G (, ) be an interval-valued fuzzy graph where [ µ, µ ] and µ, µ ] be two interval-valued fuzzy sets on a non-empty finite set V and [ E V V respectively ssume that G is a neighbourly irregular interval-valued fuzzy graph, ie the degrees of every two adjacent vertices are distinct Consider two adjacent vertices u and u with distinct degrees k, k ] and k, k ] respectively lso let, µ u) c, [ [ u where, c [ 0,] ( u) [ d ( u) c, d ( u) c ] [ k c, k c ] ( u ) [ d ( u ) c, d ( u ) c ] [ k c, k c µ ( u) c for all V c are constants Therefore, td ( td ] Clearly, td( u) td( u) Therefore for any two adjacent vertices u and u with distinct degrees, it s total degrees are also distinct, provided µ, µ are constant functions The above argument is true for every pair of adjacent vertices in G 5 Conclusions Graph theory is an extremely useful tool in solving the combinatorial problems in different areas including geometry, algebra, number theory, topology, operations research and computer science In this paper, we have described degree of a vertex, order, size and underlying crisp graph of an interval-valued fuzzy graph The necessary and sufficient conditions for an interval-valued fuzzy graph to be the regular interval-valued fuzzy graphs have been presented Size of an interval-valued fuzzy graph a relation between size and order of an interval-valued fuzzy graph have been calculated We have defined

10 Irregular Interval Valued Fuzzy Graphs irregular interval-valued fuzzy graphs, neighbourly irregular, totally and highly irregular interval-valued fuzzy graphs Some relations about the defined graphs have been proved REFERENCES M kram and W Dudec, Interval-valued fuzzy graphs, Computers and Mathematics with pplications, 6 (0), M kram, ipolar fuzzy graphs, Information Sciences, 8 (0), M kram and Davvaz, Strong intuitionistic fuzzy graphs, Filomat, 6() (0), M kram and W Dudek, Regular bipolar fuzzy graphs, Neural Computing and pplications (0) 5 K R hutani and attou, On M-Strong fuzzy graphs, Information Sciences, 55 (-), 03-09, (003) 6 K R hutani and Rosenfeld, Strong arcs in fuzzy graphs, Information Sciences, 5 (003), K R hutani, J Mordeson and Rosenfeld, On degrees of end nodes and cut nodes in fuzzy graphs, Iranian Journal of Fuzzy Systems, () (004), M Gorzalczany, method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and Systems, (987), -7 9 M Gorzalczany, n interval-valued fuzzy inference method some basic properties, Fuzzy Sets and Systems, 3 (989), J M Mendel, Uncertain Rule-ased Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, Upper Saddle River, New Jersey, (00) S Mathew and M S Sunitha, Types of arcs in a fuzzy graph, Information Sciences, 79 (009), J N Mordeson and P S Nair, Fuzzy Graphs and Hypergraphs, Physical Verlag, (000) 3 Nagoorgani and K Radha, On regular fuzzy graphs, Journal of Physical Sciences, (008), Nagoorgani and J Malarvizhi, Isomorphism properties of strong fuzzy graphs, InternJournal of lgorithms, Computing and Mathematics, () (009), Nagoorgani and P Vadivel, Relations between the parameters of independent domination and irredundance in fuzzy graph, International Journal of lgorithms, Computing and Mathematics, () (009), Nagoorgani and Latha, On irregular fuzzy graphs, pplied Mathematical Sciences, 6() (0), P S Nair and S C Cheng, Cliques and fuzzy cliques in fuzzy graphs, IFS world congress and 0 th NFIPS International Conference, 4 (00), P S Nair, Perfect and precisely perfect fuzzy graphs, Fuzzy Information Processing Society, (008) 9-9 Rosenfeld, Fuzzy graphs, in: L Zadeh, K S Fu, M Shimura (Eds), Fuzzy Sets and Their pplications, cademic Press, New York, 77-95, (975) 0 M K Roy and R iswas, I-V fuzzy relations and Sanchez s approach for medical diagnosis, Fuzzy Sets and Systems, 47 (99),

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