Certain Characterization ofm-polar Fuzzy Graphs by Level Graphs

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1 Punjab University Journal of Mathematics (ISSN ) Vol. 49(1)(2017) pp Certain Characterization ofm-polar Fuzzy Graphs by Level Graphs Muhammad Akram Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan, Gulfam Shahzadi Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan, Received: 18 September, 2016 / Accepted: 04 November, 2016 / Published online: 30 November, 2016 Abstract. Zadeh introduced the concept of fuzzy sets as a mathematical tool to deal with uncertainty, imprecision and vagueness. Since then, many higher order fuzzy sets, including intuitionistic fuzzy sets, bipolar fuzzy sets and m-polar fuzzy set, have been reported in literature to solve many real life problems, involving ambiguity and uncertainty. In this paper, we present certain characterization of m-polar fuzzy graphs by level graphs. AMS (MOS) Subject Classification Codes: 35S29, 40S70, 25U09 Key Words: Level graphs, m-polar fuzzy graphs, Characterizations. 1. INTRODUCTION Graph theory is a enjoyable playground for the research of proof techniques in discrete mathematics. There are many applications of graph theory in different fields. The world of theoretical physics discovered graph theory for its own purposes. In the study of statistical mechanics, the points represent molecules and two adjacent points indicate nearest neighbor interaction of some physical kind, like magnetic interaction or repulsion. The study of Markov chains in probability theory involves directed graphs in the sense that events are given by points and a directed line from one point to another shows a positive probability of direct succession of these two events. Job assignments problem is solved by bipartite graphs. In1994, Zhang [19] initiated the idea of bipolar fuzzy sets, which is a generalization of fuzzy set [17]. The membership degree range in a bipolar fuzzy set is [ 1,1]. In a bipolar fuzzy set, the membership degree 0 of an element means that the element is irrelevant to the corresponding property, the membership degree (0, 1] of an element indicates that the element somewhat satisfies the property, and the membership degree [ 1, 0) of an element indicates that the element somewhat satisfies the 1

2 2 Muhammad Akram and Gulfam Shahzadi implicit counter-property. The idea of m-polar fuzzy set which is an extension of a bipolar fuzzy set, studied by Chen et al. [9] and exposed that 2 polar and bipolar fuzzy set are cryptomorphic mathematical notions. The background of this concept is that multipolar information (not like the bipolar information which give two-valued logic) arise because information for a natural world are frequently from n factors (n 2). The statement Pakistan is a good country, consider as an example. The truth value of this statement may not a real number in[0,1]. Being good country may have several components: good in public transport system, good in political awareness, good in medical facilities, etc. The each component may be a real number in [0,1]. Ifnis the number of such components under consideration, then the truth value of fuzzy statement is an-tuple of real numbers in [0,1], that is, an element of[0,1] n. In 1973, Kauffmann [12] illustrated the notion of fuzzy graphs based on Zadeh s fuzzy relations [18]. The fuzzy graphs structure was described by Rosenfeld [16]. Later, Bhattacharya [8] gave some remarks on fuzzy graphs. 1994, Mordeson and Chang- Shyh [14] defined some operations on fuzzy graphs. In 2011, Akram introduced the notion of bipolar fuzzy graphs in [1]. Dudek and Talebi [10] described operations on level graphs of bipolar fuzzy graphs. Recently, Akram et al. [3 7] has discussed several new concepts, includingm-polar fuzzy graphs, certain metrics in m-polar fuzzy graphs, certain types of edge m-polar fuzzy graphs and m-polar fuzzy hypergraphs. In this research paper, we present characterization of m-polar fuzzy graphs by level graphs. 2. CHARACTERIZATION OF m-polar FUZZY GRAPHS BY LEVEL GRAPHS Definition 2.1. [9] An m-polar fuzzy set in a universe Y is a function C : Y [0,1] m. The degree of each elementa Y is written asc(a) = (P 1 oc(a),p 2 oc(a),,p m oc(a)), wherep k oc : [0,1] m [0,1] is thekth projection mapping. Note that [0,1] m (m-th power of [0,1]) is considered as a poset with the point-wise order, where m is an arbitrary ordinal number (we make an appointment that m = {n n < m} whenm > 0), is defined by a b P k (a) P k (b) for eachk m ( a,b [0,1] m ), and P k : [0,1] m [0,1] is the k-th projection mapping (k m). 1 = (1,1,,1) is the greatest value and 0 = (0,0,,0) is the smallest value in [0,1] m. Definition 2.2. [4] Let C be an m-polar fuzzy subset of a non-empty Y. An m-polar fuzzy relation on C is an m-polar fuzzy subset D of Y Y defined by the mapping D : Y Y [0,1] m such that for all a,b Y P k od(ab) inf{p k oc(a),p k oc(b)} 1 k m, where P k oc(a) denotes the k-th degree of membership of a vertex a andp k od(ab) denotes thek-th degree of membership of the edgeab. Definition 2.3. [4, 9] Anm-polar fuzzy graph is a pairg = (C,D), wherec : Y [0,1] m is an m-polar fuzzy set in Y and D : Y Y [0,1] m is an m-polar fuzzy relation ony such that P k od(ab) inf{p k oc(a),p k oc(b)}

3 Certain Characterization ofm-polar Fuzzy Graphs by Level Graphs 3 1 k m, for all a,b Y and P k od(ab) = 0 for all ab Y Y F for all k = 1,2,,m. C is called the m-polar fuzzy vertex set of G and D is called the m-polar fuzzy edge set ofg, respectively. We now definet-level set ony andf Y Y. Definition 2.4. LetC : Y [0,1] m be anm-polar fuzzy set ony. The set C t = {a Y P k oc(a) α k,1 k m} where t [0,1] m and t = (α 1,α 2,,α m ), is called the t-level set of C. Let D : Y Y [0,1] m be anm-polar fuzzy relation ony. The set D t = {ab Y Y P k od(ab) α k,1 k m} wheret [0,1] m andt = (α 1,α 2,,α m ) is calledt-level set ofd. G t = (C t,d t ) is calledt-level graph. Example 2.5. Consider a 3-polar fuzzy graph on Y = {s, t, u, v}. v(0.5,0.6,0.3) (0.5,0.6,0.3) (0.7,0.6,0.4) s(0.7,0.7,0.4) (0.5,0.6,0.3) (0.5,0.6,0.3) (0.7,0.6,0.4) u(0.9,0.6,0.6) (0.8,0.6,0.5) t(0.8,0.6,0.5) FIGURE 1. 3-polar fuzzy graphg = (C,D) Taket = (0.6,0.5,0.4). It is easy to see thatc (0.6,0.5,0.4) = {s,t,u},d (0.6,0.5,0.4) = {st,su,tu}. Clearly, the (0.6,0.5,0.4)-level graph = G (0.6,0.5,0.4) is a subgraph of crisp graphg = (Y,F). We formulate a proposition. Proposition 2.6. The level graphg t = (C t,d t ) is a crisp graph. Theorem 2.7. G is an m-polar fuzzy graph if and only if G t = (C t,d t ) is a crisp graph for eacht [0,1] m,t = (α 1,α 2,,α m ). Proof. For everyt [0,1] m,t = (α 1,α 2,,α m ). Takeab D t. ThenP k od(ab) α k,1 k m. SinceGis anm-polar fuzzy graph, it follows that α k P k od(ab) inf{p k oc(a),p k oc(b)}. This shows that α k P k oc(a), α k P k oc(b), for k = 1,2,,m, that is, a,b C t. Therefore,G t = (C t,d t ) is a graph for eacht [0,1] m, t = (α 1,α 2,,α m ). Conversely, let G t = (C t,d t ) be a graph for all t [0,1] m, t = (α 1,α 2,,α m ). For every ab Y Y, let P k od(ab) = α k, 1 k m. Then ab D t. Since G t = (C t,d t ) is a graph, we havea,b C t ; hencep k oc(a) α k, P k oc(b) α k, 1 k m. P k od(ab) = α k inf{p k oc(a),p k oc(b)}.

4 4 Muhammad Akram and Gulfam Shahzadi Thus, G is an m-polar fuzzy graph. Definition 2.8. Let G 1 = (C 1,D 1 ) and G 2 = (C 2,D 2 ) be m-polar fuzzy graphs of G 1 = (Y 1,F 1 ) and G 2 = (Y 2,F 2 ), respectively. The Cartesian product G 1 G 2 is the pair(c,d) ofm-polar fuzzy sets defined on the Cartesian productg 1 G 2 such that (i) P k oc(a 1,a 2 ) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) for all (a 1,a 2 ) Y 1 Y 2, (ii) P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 ) for all a Y 1 and for all a 2 b 2 F 2, (iii) P k od((a 1,c)(b 1,c)) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (c)) for all c Y 2 and for all a 1 b 1 F 1. Theorem 2.9. Let G 1 = (C 1,D 1 )and G 2 = (C 2,D 2 ) be m-polar fuzzy graphs of G 1 = (Y 1,F 1 ) and G 2 = (Y 2,F 2 ), respectively. Then G = (C,D) is the Cartesian product of G 1 and G 2 if and only if for each t [0,1] m, t = (α 1,α 2,,α m ) the t-level graphg t is the Cartesian product of(g 1 ) t and(g 2 ) t. Proof. For eacht [0,1] m, t = (α 1,α 2,,α m ), if(a,b) C t, then inf(p k oc 1 (a),p k oc 2 (b)) = P k oc(a,b) α k, 1 k m, so a (C 1 ) (t) and b (C 2 ) (t), that is, (a,b) (C 1 ) (t) (C 2 ) (t). Therefore, C t (C 1 ) t (C 2 ) t. Let (a,b) (C 1 ) t (C 2 ) t, then a (C 1 ) t and b (C 2 ) t. It follows that inf(p k oc 1 (a),p k oc 2 (b)) α k, 1 k m. Since (C,D) is the Cartesian product ofg 1 andg 2,P k oc(a,b) α k, that is, (a,b) C t. Therefore,(C 1 ) t (C 2 ) t C t and so (C 1 ) t (C 2 ) t = C t. We now proved t = F, wheref is the edge set of the Cartesian product(g 1 ) t and(g 2 ) t for eacht [0,1] m, t = (α 1,α 2,,α m ). Let (a 1,a 2 )(b 1,b 2 ) D t. Then, P k od((a 1,a 2 )(b 1,b 2 )) α k, 1 k m. Since (C,D) is the Cartesian product of G 1 and G 2, one of the following cases hold: (i) a 1 = b 1 anda 2 b 2 F 2. (ii) a 2 = b 2 anda 1 b 1 F 1. For the case (i), we have P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k oc 1 (a 1 ),P k od 2 (a 2 b 2 )) α k, so P k oc 1 (a 1 ) α k, P k od 2 (a 2 b 2 ) α k. It follows that a 1 = b 1 (C 1 ) t, a 2 b 2 (D 2 ) t, that is, (a 1,a 2 )(b 1,b 2 ) F. Similarly, for the case (ii), we conclude that (a 1,a 2 )(b 1,b 2 ) F. Therefore,D t F. For every(a,a 2 )(a,b 2 ) F, P k oc 1 (a) α k, P k od 2 (a 2 b 2 ) α k, 1 k m. Since (C,D) is the Cartesian product of G 1 andg 2, we have P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) α k, 1 k m. Therefore (a,a 2 )(a,b 2 ) D t. Similarly, for every (a 1,c)(b 1,c) F, we have(a 1,c)(b 1,c) D t. Therefore,F D t, and sod t = F. Conversely, suppose thatg t = (C t,d t ) is the Cartesian product of(g 1 ) t = ((C 1 ) t,(d 1 ) t ) and(g 2 ) t = ((C 2 ) t,(d 2 ) t ) for allt [0,1] m,t = (α 1,α 2,,α m ). Let inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) = α k, 1 k m for some (a 1,a 2 ) Y 1 Y 2. Thena 1 (C 1 ) t anda 2 (C 2 ) t. By hypothesis,(a 1,a 2 ) C t, hence P k oc(a 1,a 2 ) α k = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 ))

5 Certain Characterization ofm-polar Fuzzy Graphs by Level Graphs 5 Take P k oc(a 1,a 2 ) = β k,1 k m, then (a 1,a 2 ) C t where t [0,1] m, t = (β 1,β 2,,β m ). Since (C t,d t ) is the Cartesian product of ((C 1 ) t,(d 1 ) t ) and((c 2 ) t,(d 2 ) t ), thena 1 (C 1 ) t anda 2 (C 2 ) t. Hence, It follows that Therefore, P k oc 1 (a 1 ) β k,p k oc 2 (a 2 ) β k inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) P k oc(a 1,a 2 ) P k oc(a 1,a 2 ) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) forall(a 1,a 2 ) Y 1 Y 2. Similarly, for everya Y 1 and everya 2 b 2 F 2, let inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) = α k, P k od((a,a 1 )(a,b 2 )) = β k,1 k m. Then we have P k oc 1 (a) α k, P k od 2 (a 2 b 2 ) α k, that is, a (C 1 ) t, a 2 b 2 (D 2 ) t, t = (α 1,α 2,,α m ) and(a,a 2 )(a,b 2 ) D t, t = (β 1,β 2,,β m ). Since (C t,d t )(resp.(c t,d t )) is the Cartesian product of((c 1 ) t,(d 1 ) t ) and((c 2 ) t,(d 2 ) t ) (resp.(c 1 ) t,(d 1 ) t ) and ((C 2 ) t,(d 2 ) t ) we have (a,a 2 )(a,b 2 ) D t, a (C 1 ) t and a 2 b 2 (D 2 ) t, which implies P k oc 1 (a) β k, P k od 2 (a 2 b 2 ) β k. It follows that P k od((a,a 2 )(a,b 2 )) α k = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )), Therefore, inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) β k = P k od((a,a 2 )(a,b 2 )). P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) for all a Y 1 anda 2 b 2 F 2. Similarly, we can show that P k od((a 1,c)(b 1,c)) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (c)) for all c Y 2 anda 1 b 1 F 1. This completes the proof. Definition LetG 1 andg 2 bem-polar fuzzy graphs ofg 1 andg 2, respectively. The composition G 1 [G 2 ] is the pair (C,D) of m-polar fuzzy sets defined on the compositiong 1[G 2] such that (i) P k oc(a 1,a 2 ) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) for all (a 1,a 2 ) Y 1 Y 2, (ii) P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) for alla Y 1 and for all a 2 b 2 F 2, (iii) P k od((a 1,c)(b 1,c)) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (c)) for all c Y 2 and for all a 1 b 1 F 1, (iv) P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (a 2 ),P k oc 2 (b 2 )) for all a 2,b 2 Y 2, wherea 2 b 2 and for all a 1 b 1 F 1. Theorem Let G 1 and G 2 be m-polar fuzzy graphs of G 1 and G 2, respectively. Then G is the composition of G 1 and G 2 if and only if for each t [0,1] m, t = (α 1,α 2,,α m ) thet-level graphg t is the composition of (G 1 ) t and(g 2 ) t.

6 6 Muhammad Akram and Gulfam Shahzadi Proof. By the definition of G 1 [G 2 ] and in the same way as in the proof of Theorem 2.9, we have C t = (C 1 ) t (C 2 ) t. We prove D t = F, where F is the edge set of the composition (G 1 ) t [(G 2 ) t ] for all t [0,1] m, t = (α 1,α 2,,α m ). Let (a 1,a 2 )(b 1,b 2 ) D t. Then P k od((a 1,a 2 )(b 1,b 2 )) α k, 1 k m. Since G is the compositiong 1 [G 2 ], one of the following cases hold: (i) a 1 = b 1 anda 2 b 2 F 2. (ii) a 2 = b 2 anda 1 b 1 F 1. (iii) a 2 b 2 anda 1 b 1 F 1. For the cases (i) and (ii), similarly as in the cases (i) and (ii) in the proof of Theorem 2.9, we obtain(a 1,a 2 )(b 1,b 2 ) F. For the case (iii), we have P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (a 2 ),P k oc 2 (b 2 )) α k Thus, P k oc 2 (a 2 ) α k, P k oc 2 (b 2 ) α k, P k od 1 (a 1 b 1 ) α k, 1 k m. It follows thata 2,b 2 (C 2 ) t anda 1 b 1 (D 1 ) t, that is,(a 1,a 2 )(b 1,b 2 ) F. Therefore, D t F. For every (a,a 2 )(a,b 2 ) F,P k oc 1 (a) α k,p k od 2 (a 2 b 2 ) α k, 1 k m. SinceG = (C,D) is the compositiong 1 [G 2 ], we have P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) α k, 1 k m. Therefore, (a,a 2 )(a,b 2 ) D t. Similarly, for every (a 1,c)(b 1,c) F, we have(a 1,c)(b 1,c) D t. For every(a 1,a 2 )(b 1,b 2 ) F wherea 2 b 2, a 1 b 1, P k od 1 (a 1 b 1 ) α k, P k oc 2 (a 2 ) α k, P k oc 2 (b 2 ) α k, 1 k m. Since G is the compositiong 1 [G 2 ], we have P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (a 2 ),P k oc 2 (b 2 )) α k 1 k m. Thus,(a 1,a 2 )(b 1,b 2 ) D t. Therefore,F D t, and so F = D t. Conversely, suppose that G t = (C t,d t ), where t [0,1] m,t = (α 1,α 2,,α m ) is the composition of (G 1 ) t = ((C 1 ) t,(d 1 ) t ) and (G 2 ) t = ((C 2 ) t,(d 2 ) t ). By the definition of the composition and the proof of Theorem 2.9, we have (i) P k oc(a 1,a 2 ) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) for all (a 1,a 2 ) Y 1 Y 2, (ii) P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) for alla Y 1 and for all a 2 b 2 F 2, (iii) P k od((a 1,c)(b 1,c)) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (c)) for all c Y 2 and for all a 1 b 1 F 1. Similarly, by using same arguments as in the proof of Theorem 2.9, we obtain P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (a 2 ),P k oc 2 (b 2 )) for all a 2,b 2 Y 2 (a 2 b 2 ) and for all a 1 b 1 F 1. This completes the proof. Definition LetG 1 andg 2 bem-polar fuzzy graphs ofg 1 andg 2, respectively. The uniong 1 G 2 is defined as the pair(c,d) ofm-polar fuzzy sets determined on the union of graphsg 1 andg 2 such that P k oc 1 (a) if a Y 1 anda Y 2, (i) P k oc(a) = P k oc 2 (a) if a Y 2 anda Y 1, sup(p k oc 1 (a),p k oc 2 (a)) if a Y 1 Y 2. (ii) P k od(ab) = P k od 1 (ab) if ab F 1 and ab F 2, P k od 2 (ab) if ab F 2 and ab F 1, sup(p k od 1 (ab),p k od 2 (ab)) if ab F 1 F 2.

7 Certain Characterization ofm-polar Fuzzy Graphs by Level Graphs 7 Theorem Let G 1 andg 2 bem-polar fuzzy graphs of G 1 andg 2, respectively, andy 1 Y 2 =. Then G is the union of G 1 andg 2 if and only if eacht-level graph G t is the union of(g 1 ) t and(g 2 ) t. Proof. We show thatc t = (C 1 ) t (C 2 ) t for eacht [0,1] m,t = (α 1,α 2,,α m ). Let a C t, then a Y 1 \ Y 2 or a Y 2 \ Y 1. If a Y 1 \ Y 2, then P k oc 1 (a) = P k oc(a) α k,1 k m which implies a (C 1 ) t. Analogously a Y 2 \ Y 1 impliesa (C 2 ) t. Therefore,a (C 1 ) t (C 2 ) t, and so C t (C 1 ) t (C 2 ) t. Now let a (C 1 ) t (C 2 ) t. Thena (C 1 ) t, a (C 2 ) t ora (C 2 ) t, a (C 1 ) t. For the first case, we have P k oc 1 (a) = P k oc(a) α k,1 k m which implies a C t. For the second case, we have P k oc 2 (a) = P k oc(a) α k,1 k m. Hence a C t. Consequently,(C 1 ) t (C 2 ) t C t. To prove that D t = (D 1 ) t (D 2 ) t, for all t [0,1] m,t = (α 1,α 2,,α m ), consider ab D t. Then ab F 1 \ F 2 or ab F 2 \ F 1. For ab F 1 \ F 2 we have P k od 1 (ab) = P k od(ab) α k,1 k m. Thus ab (D 1 ) t. Similarly ab F 2 \ F 1 gives ab (D 2 ) t. Therefore D t (D 1 ) t (D 2 ) t. If ab (D 1 ) t (D 2 ) t, then ab (D 1 ) t \ (D 2 ) t or ab (D 2 ) t \ (D 1 ) t. For the first case P k od(ab) = P k od 1 (ab) α k,1 k m, hence ab D t. In the second case we obtainab D t. Therefore,(D 1 ) t (D 2 ) t D t. Conversely, let for all t [0,1] m,t = (α 1,α 2,,α m ) the level graph G t = (C t,d t ) be the union of (G 1 ) t = ((C 1 ) t,(d 1 ) t ) and (G 2 ) t = ((C 2 ) t,(d 2 ) t ). Let a Y 1,P k oc 1 (a) = α k,p k oc(a) = β k,1 k m, Then a (C 1 ) t where t [0,1] m,t = (α 1,α 2,,α m ) and a C t where t [0,1] m,t = (β 1,β 2,,β m ). But by the hypothesisa (C 1 ) t anda C t. Thus, P k oc 1 (a) β k,p k oc(a) α k,1 k m. Therefore,P k oc 1 (a) P k oc(a) andp k oc 1 (a) P k oc(a). Hence P k oc 1 (a) = P k oc(a). Similarly, for every a Y 2, we get P k oc 2 (a) = P k oc(a). Thus we conclude that { Pk oc(a) = P (i) k oc 1 (a) if a Y 1, P k oc(a) = P k oc 2 (a) if a Y 2. By a similar method as above, we obtain { Pk od(ab) = P (ii) k od 1 (ab) if ab F 1, P k od(ab) = P k od 2 (ab) if ab F 2. This completes the proof. Definition LetG 1 andg 2 bem-polar fuzzy graphs ofg 1 andg 2, respectively. The joing 1 +G 2 is the pair(c,d) ofm-polar fuzzy sets defined on the joing 1 +G 2 such that (i) P k oc(a) = (ii) P k od(ab) = P k oc 1 (a) if a Y 1 anda Y 2, P k oc 2 (a) if a Y 2 anda Y 1, sup(p k oc 1 (a),p k oc 2 (a)) if a Y 1 Y 2. P k od 1 (ab) if ab F 1 and ab F 2, P k od 2 (ab) if ab F 2 and ab F 1, sup(p k od 1 (ab),p k od 2 (ab)) if ab F 1 F 2, inf(p k oc 1 (a),p k oc 2 (b)) if ab F.

8 8 Muhammad Akram and Gulfam Shahzadi Theorem Let G 1 andg 2 bem-polar fuzzy graphs of G 1 andg 2, respectively, and Y 1 Y 2 =. Then G is the join of G 1 and G 2 if and only if each t-level graph G t is the join of(g 1 ) t and(g 2 ) t. Proof. By the definition of union and the proof of Theorem 2.13,C t = (C 1 ) t (C 2 ) t, for all t [0,1] m,t = (α 1,α 2,,α m ). We show that D t = (D 1 ) t (D 2 ) t F t for all t [0,1] m,t = (α 1,α 2,,α m ), wheref t is the set of all edges joining the vertices of(c 1 ) t and(c 2 ) t. From the proof of Theorem 2.13, it follows that (D 1 ) t (D 2 ) t D t. If ab F t, thenp k oc 1 (a) α k,p k oc 2 (b) α k,1 k m. Hence P k od(ab) = inf(p k oc 1 (a),p k oc 2 (b)) α k It follows thatab D t. Therefore,(D 1 ) t (D 2 ) t F t D t. For everyab D t, if ab F 1 F 2, thenab (D 1 ) t (D 2 ) t, by the proof of Theorem Ifa Y 1 and b Y 2, then inf(p k oc 1 (a),p k oc 2 (b)) = P k od(ab) α k, soa (C 1 ) t andb (C 2 ) t. Thusab F t. Therefore,D t (D 1 ) t (D 2 ) t F t. Conversely, let each level graphg t = (C t,d t ) be the join of(g 1 ) t = ((C 1 ) t,(d 1 ) t ) and(g 2 ) t = ((C 2 ) t,(d 2 ) t ). From the proof of the Theorem 2.13, we have { Pk oc(a) = P (i) k oc 1 (a) if a Y 1, { P k oc(a) = P k oc 2 (a) if a Y 2. Pk od(ab) = P (ii) k od 1 (ab) if ab F 1, P k od(ab) = P k od 2 (ab) if ab F 2. let a Y 1,b Y 2,inf(P k oc 1 (a),p k oc 2 (b)) = α k,p k od(ab) = β k. Then a (C 1 ) t,b (C 2 ) t where t [0,1] m,t = (α 1,α 2,,α m ) and ab D t where t [0,1] m,t = (β 1,β 2,,β m ). It follows that ab D t,a (C 1 ) t and b (C 2 ) t. So,P k od(ab) α k,p k oc 1 (a) β k andp k oc 2 (b) β k. Therefore, Thus, P k od(ab) α k = inf(p k oc 1 (a),p k oc 2 (b)) β k = P k od(ab). P k od(ab) = inf(p k oc 1 (a),p k oc 2 (b)). Definition LetG 1 andg 2 bem-polar fuzzy graphs ofg 1 andg 2, respectively. The cross product G 1 G 2 is the pair (C,D) of m-polar fuzzy sets defined on the cross productg 1 G 2 such that (i) P k oc(a 1,a 2 ) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) for all (a 1,a 2 ) Y 1 Y 2, (ii) P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) for all a 1 b 1 F 1 and for alla 2 b 2 F 2. Theorem Let G 1 andg 2 bem-polar fuzzy graphs of G 1 andg 2, respectively. Then G = (C,D) is the cross product of G 1 and G 2 if and only if each level graph G t is the cross product of(g 1 ) t and(g 2 ) t. Proof. By the definition of the Cartesian product and the proof of Theorem 2.9, we havec t = (C 1 ) t (C 2 ) t, for all t [0,1] m,t = (α 1,α 2,,α m ). We show that D t = {(a 1,a 2 )(b 1,b 2 ) a 1 b 1 (D 1 ) t, a 2 b 2 (D 2 ) t }

9 Certain Characterization ofm-polar Fuzzy Graphs by Level Graphs 9 for all t [0,1] m,t = (α 1,α 2,,α m ). Infact, if(a 1,a 2 )(b 1,b 2 ) D t, then P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) α k so P k od 1 (a 1 b 1 ) α k and P k od 2 (a 2 b 2 ) α k,1 k m. So, a 1 b 1 (D 1 ) t and a 2 b 2 (D 2 ) t. Now ifa 1 b 1 (D 1 ) t anda 2 b 2 (D 2 ) t, thenp k od 1 (a 1 b 1 ) α k and P k od 2 (a 2 b 2 ) α k,1 k m. It follows that P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) α k SinceG = (C,D) is the cross product ofg 1 G 2. Therefore,(a 1,a 2 )(b 1,b 2 ) D t. Conversely, let each t-level graph G t = (C t,d t ) be the cross product of (G 1 ) t = ((C 1 ) t,(d 1 ) t ) and (G 2 ) t = ((C 2 ) t,(d 2 ) t ). In view of the fact that the cross product (C t,d t ) has the same vertex set as the Cartesian product of ((C 1 ) t,(d 1 ) t ) and ((C 2 ) t,(d 2 ) t ), and by the proof of Theorem 2.9, we have P k oc((a 1,a 2 )) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) forall(a 1,a 2 ) Y 1 Y 2. Let inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) = α k and P k od((a 1,a 2 )(b 1,b 2 )) = β k,1 k m for a 1 b 1 F 1, a 2 b 2 F 2. Then P k od 1 (a 1 b 1 ) α k,p k od 2 (a 2 b 2 ) α k and (a 1,a 2 )(b 1,b 2 ) D t where t [o,1] m,t = (β 1,β 2,,β m ), hence a 1 b 1 (D 1 ) t, a 2 b 2 (D 2 ) t, where t [o,1] m,t = (α 1,α 2,,α m ) and consequently a 1 b 1 (D 1 ) t, a 2 b 2 (D 2 ) t, since D t = {(a 1,a 2 )(b 1,b 2 ) a 1 b 1 (D 1 ) t, a 2 b 2 (D 2 ) t }. It follows that (a 1,a 2 )(b 1,b 2 ) D t, P k od 1 (a 1 b 1 ) β k,p k od 2 (a 2 b 2 ) β k,1 k m. Therefore, P k od((a 1,a 2 )(b 1,b 2 )) = β k inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) = α k P k od((a 1,a 2 )(b 1,b 2 )). Hence P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )). This completes the proof. Definition LetG 1 andg 2 bem-polar fuzzy graphs. The lexicographic product G 1 G 2 is the pair(c,d) ofm-polar fuzzy sets defined on the lexicographic product G 1 G 2 such that (i) P k oc(a 1,a 2 ) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) for all (a 1,a 2 ) Y 1 Y 2, (ii) P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) for alla Y 1 and for all a 2 b 2 F 2, (iii) P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) for all a 1 b 1 F 1 and for alla 2 b 2 F 2. Theorem LetG 1 andg 2 bem-polar fuzzy graphs. ThenGis the lexicographic product of G 1 and G 2 if and only if G t = (G 1 ) t (G 2 ) t for all t [0,1] m,t = (α 1,α 2,,α m ). Proof. By the definition of Cartesian product G 1 G 2 and the proof of Theorem 2.9, we have C t = (C 1 ) t (C 2 ) t for all t [0,1] m,t = (α 1,α 2,,α m ). We show that D t = F t F t for all t [0,1]m,t = (α 1,α 2,,α m ), where F t = {(a,a 2 )(a,b 2 ) a Y 1, a 2 b 2 (D 2 ) t } is the subset of the edge set of the cross product(g 1 ) t (G 2 ) t, and F t = {(a 1,a 2 )(b 1,b 2 ) a 1 b 1 (D 1 ) t, a 2 b 2 (D 2 ) t } is the edge set of the cross product (G 1 ) t (G 2 ) t. For every (a 1,a 2 )(b 1,b 2 ) D t, a 1 = b 1, a 2 b 2 F 2 or a 1 b 1 F 1, a 2 b 2 F 2. If a 1 = b 1, a 2 b 2 F 2, then(a 1,a 2 )(b 1,b 2 ) F t, by the definition of the Cartesian product and the proof of Theorem 2.9. If a 1 b 1 F 1, a 2 b 2 F 2, then (a 1,a 2 )(b 1,b 2 ) F t, by the definition

10 10 Muhammad Akram and Gulfam Shahzadi of cross product and the proof of Theorem Therefore,D t F t F t. From the definition of the Cartesian product and the proof of Theorem 2.9, we conclude that F t D t, and also from the definition of cross product and the proof of Theorem 2.17, we obtainf t D t. Therefore,F t F t D t. Conversely, letg t = (C t,d t ) = (G 1 ) t (G 2 ) t for allt [0,1] m,t = (α 1,α 2,,α m ). We know that(g 1 ) t (G 2 ) t has the same vertex set as the Cartesian product(g 1 ) t (G 2 ) t. Now by the proof of Theorem 2.9, we have P k oc((a 1,a 2 )) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) forall(a 1,a 2 ) Y 1 Y 2. Let for a Y 1 and a 2 b 2 F 2 will be inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) = α k and P k od((a,a 2 )(a,b 2 )) = β k,1 k m. Then, in view of the definitions of the Cartesian product and lexicographic product, we have (a,a 2 )(a,b 2 ) (D 1 ) t (D 2 ) t (a,a 2 )(a,b 2 ) (D 1 ) t (D 2 ) t, (a,a 2 )(a,b 2 ) (D 1 ) t (D 2 ) t (a,a 2 )(a,b 2 ) (D 1 ) t (D 2 ) t. From this, by the same way as in the proof of Theorem 2.9, we conclude P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )). Now let P k od((a 1,a 2 )(b 1,b 2 )) = α k, inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) = β k,1 k m fora 1 b 1 F 1 anda 2 b 2 F 2. Then in view of the definitions of cross product and the lexicographic product, we have (a 1,a 2 )(b 1,b 2 ) (D 1 ) t (D 2 ) t (a 1,a 2 )(b 1,b 2 ) (D 1 ) t (D 2 ) t, (a 1,a 2 )(b 1,b 2 ) (D 1 ) t (D 2 ) t (a 1,a 2 )(b 1,b 2 ) (D 1 ) t (D 2 ) t. By the same way as in the proof of Theorem 2.17, we can conclude P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )), which completes the proof. Proposition Let G 1 and G 2 be m-polar fuzzy graphs of G 1 = (Y 1,F 1 ) and G 2 = (Y 2,F 2 ), respectively, such that Y 1 = Y 2, C 1 = C 2 and F 1 F 2 =. Then G = (C,D) is the union of G 1 and G 2 if and only if G t is the union of (G 1 ) t and (G 2 ) t for all t [0,1] m,t = (α 1,α 2,,α m ). Proof. Let G = (C,D) be the union of m-polar fuzzy graphs G 1 and G 2. Then by the definition of the union and the fact that Y 1 = Y 2, C 1 = C 2, we have C = C 1 = C 2, hence C t = (C 1 ) t (C 2 ) t. We now show that D t = (D 1 ) t (D 2 ) t for all t [0,1] m,t = (α 1,α 2,,α m ). For every ab (D 1 ) t we have P k od(ab) = P k od 1 (ab) α k,1 k m, hence ab D t. Therefore, (D 1 ) t D t. Similarly we obtain (D 2 ) t D t. Thus, (D 1 ) t (D 2 ) t D t. For every ab D t, ab F 1 or ab F 2. If ab F 1, P k od 1 (ab) = P k od(ab) α k,1 k m and hence ab (D 1 ) t. Ifab F 2, we haveab (D 2 ) t. Therefore,D t (D 1 ) t (D 2 ) t. Conversely, suppose that the t-level graph G t = (C t,d t ) be the union of (G 1 ) t = ((C 1 ) t,(d 1 ) t ) and(g 2 ) t = ((C 2 ) t,(d 2 ) t ). LetP k oc(a) = α k,p k oc 1 (a) = β k,1 k m for somea Y 1 = Y 2. Thena C t wheret [o,1] m,t = (α 1,α 2,,α m ) and a (C 1 ) t where t [0,1] m,t = (β 1,β 2,,β m ), so a (C 1 ) t and a C t, because C t = (C 1 ) t and C t = (C 1 ) t. It follows that P k oc 1 (a) α k, and P k oc(a) β k,1 k m. Therefore,P k oc 1 (a) P k oc(a)and P k oc(a)

11 Certain Characterization ofm-polar Fuzzy Graphs by Level Graphs 11 P k oc 1 (a). So, P k oc(a) = P k oc 1 (a). Since C 1 = C 2, Y 1 = Y 2, then C = C 1 = C 1 C 2. By a similar method, we conclude that { Pk od(ab) = P (1) k od 1 (ab) if ab F 1, P k od(ab) = P k od 2 (ab) if ab F 2. This completes the proof. Definition LetG 1 andg 2 bem-polar fuzzy graphs ofg 1 andg 2, respectively. The strong product G 1 G 2 is the pair (C,D) of m-polar fuzzy sets defined on the strong productg 1 G 2 such that (i) P k oc(a 1,a 2 ) = inf(p k oc 1 (a 1 ),P k oc 2 (a 2 )) for all (a 1,a 2 ) Y 1 Y 2, (ii) P k od((a,a 2 )(a,b 2 )) = inf(p k oc 1 (a),p k od 2 (a 2 b 2 )) for alla Y 1 and for all a 2 b 2 F 2, (iii) P k od((a 1,c)(b 1,c)) = inf(p k od 1 (a 1 b 1 ),P k oc 2 (c)) for all c Y 2 and for all a 1 b 1 F 1, (iv) P k od((a 1,a 2 )(b 1,b 2 )) = inf(p k od 1 (a 1 b 1 ),P k od 2 (a 2 b 2 )) for all a 1 b 1 F 1 and for alla 2 b 2 F 2. We state the following Theorem without its proof. Theorem Let G 1 andg 2 bem-polar fuzzy graphs of G 1 andg 2, respectively. Then G is the strong product of G 1 and G 2 if and only if G t, where t [0,1] m,t = (α 1,α 2,,α m ), is the strong product of(g 1 ) t and(g 2 ) t. 3. CONCLUSION An m-polar fuzzy set is an extension of a bipolar fuzzy set. An m-polar fuzzy model is useful for multi-polar information, multi-agent, multi-attribute and multiobject network models which gives more precision, flexibility, and comparability to the system as compared to the classical, fuzzy and bipolar fuzzy models. In this research article, we have presented certain characterization of m-polar fuzzy graphs by level graphs. We have aim to extend our work to (1) single-valued neutrosophic soft graph structures, (2) single-valued neutrosophic rough fuzzy graph structures, (3) single-valued neutrosophic rough fuzzy soft graph structures, and (4) single-valued neutrosophic fuzzy soft graph structures. REFERENCES [1] M. Akram, Bipolar fuzzy graphs, Information Sciences, 181, (2011) [2] M. Akram, Bipolar fuzzy graphs with applications, Knowledge-Based Systems, 9, (2013) 1 8. [3] M. Akram and A. Adeel, m-polar fuzzy labeling graphs with application, Mathematics in Computer Science, 10, (2016) [4] M. Akram and N. Waseem, Certain metrices in m polar fuzzy graphs, New Mathematics and Natural Computation, 12, (2016) [5] M. Akram, N. Waseem and W. A. Dudek, Certain types of edge m polar fuzzy graphs, Iranian Journal of Fuzzy Systems, (2016) (in press). [6] M. Akram and H.R Younas, Certain types of irregular m polar fuzzy graphs, Journal of Applied Mathematics and Computing, DOI: /s , (2016). [7] M. Akram and M. Sarwar, Novel application of m polar fuzzy hypergraphs, Journal of intelligent and fuzzy systems, 31, (2016), DOI: /JIFS [8] P. Bhattacharya, Remark on fuzzy graphs, Pattern Recognition Letter 6, (1987)

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