On Direct Sum of Two Fuzzy Graphs

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1 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 ISSN On Direct Sum of Two Fuzzy raphs Dr. K. Raha *, Mr.S. Arumugam ** * P. & Research Department of Mathematics, Periyar E.V.R. College, Tiruchirapalli-6003 ** ovt. High School, Thinnanur, Tiruchirapalli Abstract- In this paper, the irect sum of two fuzzy graphs an is efine. It is prove that when two fuzzy graphs are effective then their irect sum nee not be effective. The egrees of the vertices in the irect sum of two fuzzy graphs an in terms of egrees of the vertices in the fuzzy graphs an are obtaine. The lower an upper truncations of the irect sum of two fuzzy graphs are obtaine. The regular property an connecteness of the irect sum of two fuzzy graphs are also stuie. Inex Terms- Fuzzy raph, Direct Sum, Effective Fuzzy raph, Regular Fuzzy raph, Connecteness, Upper an Lower Truncations. F I. INTRODUCTION uzzy graph theory was introuce by Azriel Rosenfel in 975. The properties of fuzzy graphs have been stuie by Azriel Rosenfel[7]. Later on, Bhattacharya[6] gave some remarks on fuzzy graphs, an some operations on fuzzy graphs were introuce by Moreson.J.N. an Peng.C.S.[3]. The conjunction of two fuzzy graphs was efine by Nagoor ani.a an Raha.K.[4]. In this paper, the irect sum of two fuzzy graphs is efine. The egree of a vertex in the irect sum of two fuzzy graphs an in terms of egrees of the vertices in the fuzzy graphs an is obtaine. This has been illustrate through some examples. The regular properties of the irect sum of two fuzzy graphs have been stuie. It is illustrate that the irect sum of two connecte fuzzy graphs an nee not be a connecte fuzzy graph. A fuzzy graph is a pair of functions :(σ, μ) where σ is a fuzzy subset of a non empty set V an μ is a symmetric fuzzy relation on σ. The unerlying crisp graph of :(σ, μ) is enote by *(V, E) where E V V. Let :(σ, μ) be a fuzzy graph. The unerlying crisp graph of :(σ, μ) is enote by *:(V, E) where E V V. A fuzzy graph is an effective fuzzy graph if μ(uv) = σ(u) σ(v) for all uv E. is complete if μ(uv) = σ(u) σ(v) for all u,v V. Therefore is a complete fuzzy graph if an only if is an effective fuzzy graph an * is complete. (σ, μ ) is a spanning fuzzy subgraph of (σ,μ) if σ =σ an μ μ, that is, if σ (u) = σ (u) for every u V an μ (e) μ(e) for every e E. (u) (uv) The egree of a vertex u is efine as uv. Since (uv) >0 for uv E an (uv) =0 for uv E, this can be (u) (uv) expresse as uve. Let :(σ, μ) be a fuzzy graph on *:(V,E). If (v)=k for all v V, that is, if each vertex has same egree k, then is sai to be a regular fuzzy graph of egree k or a k-regular fuzzy graph. Let :(σ, μ) be a fuzzy graph on *. The total egree of a t (u) (uv) (u) vertex uv is efine by uve = (u) + σ(u). If each vertex of has the same total egree k, then is sai to be a totally regular fuzzy graph of total egree k or a k- totally regular fuzzy graph. The lower an upper truncations[] of σ at level t, 0 < t, are the fuzzy subsets σ (t) an σ (t) efine respectively by, t t (u),if u t,if u t t (u) an (u) t t 0,if u (u),if u. Let :(σ,μ) be a fuzzy graph with unerlying crisp graph *:(V,E). Take V (t) = σ t, E (t) = μ t. Then (t) :(σ (t),μ (t) ) is a fuzzy graph with unerlying crisp graph (t) *:(V (t), E (t) ). This is calle the lower truncation[5] of the fuzzy graph at level t. Here V (t) an E (t) may be proper subsets of V an E respectively. Take V (t) = V, E (t) = E. Then (t) :(σ (t), μ (t) ) is a fuzzy graph with unerlying crisp graph (t) *:(V (t),e (t) ). This is calle the upper truncation of the fuzzy graph at level t. II. DIRECT SUM Let :(σ,μ ) an :(σ,μ ) enote two fuzzy graphs with unerlying crisp graphs *:(V,E ) an *:(V,E ) respectively. Let V = V V an let E = {uv / u,v V; uv E or uv E but not both }. Define :(σ, μ) by (u),if u V (uv),if uv E (u) (u),if u V an (uv) (uv),if uv E (u) (u),if u V V Then if uv E, μ(uv) = μ (uv) σ (u) σ (v) σ(u) σ(v), if uv E, μ(uv) = μ (uv) σ (u) σ (v) σ(u) σ(v). Therefore (σ, μ) efines a fuzzy graph. This is calle the irect sum of two fuzzy graphs.

2 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 ISSN Example The following Fig. gives an example of the irect sum of two fuzzy graphs which have istinct ege sets. Figure : Direct sum of two fuzzy graphs with isjoint ege sets.example The following Fig. gives an example of the irect sum of two fuzzy graphs in which the ege sets are not isjoint. Figure : Direct sum of two fuzzy graphs with non-isjoint ege sets.3remark If an are two effective fuzzy graphs, their irect sum nee not be an effective fuzzy graph which can be seen from the example in Figure 3. Figure 3: Direct sum of two effective fuzzy graphs.4theorem If an are two effective fuzzy graphs such that no ege of has both ens in V V an every ege uv of with one en u V V an uv E ( or E ) is such that σ (u) σ (v) [ or σ (u) σ (v)], then is an effective fuzzy graph. Proof: Let uv be an ege of. We have two cases to consier. Case : u, v V V. Then u, v V or V but not both. Suppose that u, v V. Then uv E. Therefore σ(u) = σ (u), σ(v) = σ (v) an μ(uv) = μ (uv). Also since is an effective fuzzy graph, μ(uv) = μ (uv) = σ (u) σ (v) = σ(u) σ(v). The proof is similar if u, v V.

3 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 3 ISSN Case : u V V, v V V. (or vice versa). Without loss of generality, assume that v V. Then σ(v) = σ (v). By hypothesis, σ (u) σ (v). Now σ(u) = σ (u) σ (u) σ (u) σ (v) = σ(v). So σ(u) σ(v) = σ(v). Hence μ(uv) = μ (uv) = σ (u) σ (v) = σ (v) = σ(v) = σ(u) σ(v). Therefore is an effective fuzzy graph. III. TRUNCATIONS OF THE DIRECT SUM OF TWO FUZZY RAPHS 3.Theorem ( ) (t) is a spanning fuzzy sub graph of (t) (t). Proof: First we prove that (σ σ ) (t) = σ (t) σ (t). It follows from the efinitions that if u V i, (σ σ ) (t) (u) = σ i(t) (u) = (σ (t) σ (t) )(u), i =,. Let u V V. Without loss of generality, assume that σ (u) σ (u). Then (σ σ )(u) = σ (u) gives (σ σ ) (t) (u) = σ (t) (u)... () Now we claim that σ (u) σ (u) implies σ (t) (u) σ (t) (u). If t σ (u) σ (u), then σ (t) (u) = σ (u) σ (u) = σ (t) (u). If σ (u) < t σ (u), then σ (t) (u) = 0 < σ (u) = σ (t) (u). If σ (u) σ (u) < t, then σ (t) (u) = 0 = σ (t) (u). Hence σ (t) (u) σ (t) (u). Therefore (σ (t) σ (t) )(u) = σ (t) (u)... () From () & (), (σ σ ) (t) (u) = (σ (t) σ (t) ) (u). Hence (σ σ ) (t) = σ (t) σ (t). Next we prove that (μ μ ) (t) μ (t) μ (t). For this, we consier the following three cases: Case : uv E E with either μ (uv) t or μ (uv) t but not both. Suppose that μ (uv) t. Then μ (uv) < t. So μ (t) (uv) = μ(uv) an μ (t) (uv) = 0. Hence the ege uv will be in (t) (t) with (μ (t) μ (t) )(uv) = μ(uv). Since uv E E, (μ μ ) (uv) = 0 (μ μ ) (t) (uv) = 0. Therefore (μ μ ) (t) (uv) < (μ (t) μ (t) )(uv). The proof is similar if μ (uv) t. Case : uv E E with either μ (uv) < t, μ (uv) < t or μ (uv) t, μ (uv) t. Since uv E E, (μ μ ) (t) (uv) = 0. If μ i (uv) < t, then μ i(t) (uv) = 0, i =,. So (μ (t) μ (t) ) (uv) = 0. If μ i (uv) t, then μ i(t) (uv) = μ i (uv) > 0, i =,. So (μ (t) μ (t) ) (uv) = 0. Hence (μ μ ) (t) = μ (t) μ (t). Case 3: uv E or uv E but not both. If uv E i, (μ μ )(uv) = μ i (uv), i =,. Hence (μ μ ) (t) (uv) = μ i(t) (uv) = (μ (t) μ (t) ) (uv). From the above three cases, we get (μ μ ) (t) μ (t) μ (t). Hence ( ) (t) is a spanning fuzzy sub graph of (t) (t). 3.Remark: From the proof of the above theorem, if for any uv E E, we have either μ (uv) < t, μ (uv) < t or μ (uv) t, μ (uv) t, then ( ) (t) = (t) (t). 3.3Theorem: ( ) (t) = (t) (t). Proof: First we prove that (σ σ ) (t) = σ (t) σ (t). It follows from the efinitions that if u Vi, (σ σ ) (t) (u) = σ i(t) (u) = (σ (t) σ (t) )(u), i =,. Let u V V. Without loss of generality, assume that σ (u) σ (u). Then proceeing as in the previous Theorem, that (σ σ ) (t) (u) = (σ (t) σ (t) )(u). Hence (σ σ ) (t) (u) = (σ (t) σ (t) )(u). Next we prove that (μ μ ) (t) = μ (t) μ (t). For this, we consier the following two cases: Case : uv E E. Then (μ μ ) (uv) = 0 (μ μ ) (t) (uv) = 0. Since μ i (uv) > 0, μ i(t) (uv) > 0, i =,. Therefore the ege uv will not be in (t) (t). So (μ (t) μ (t) )(uv) = 0. Hence (μ μ ) (t) (uv) = (μ (t) μ (t) )(uv). Case : uv E or uv E but not both. If uv E i, (μ μ ) (t) (uv) =μ i (uv), i =,.

4 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 4 ISSN Hence (μ μ ) (t) (uv) = μ i (t)(uv) = (μ (t) μ (t) )(uv). From the above two cases, we get (μ μ ) (t) (uv) = (μ (t) μ (t) )(uv). Hence : ( ) (t) = (t) (t). IV. DEREE OF A VERTEX IN THE DIRECT SUM In this section we fin the egrees of the vertices in the irect sum of two fuzzy graphs an in terms of egrees of the vertices in the fuzzy graphs an. 4.Theorem: The egree of a vertex in : (, ) in terms of the egrees of the vertices in (, ) an (, ) is given by, (u), if u V V (u), if u V V (u) (u) (u), if u V V an E E [ (u) (u)] [ (uv) (uv)], if u V V an E E uvee Proof: For any vertex in the irect sum : (, ) we have three cases to consier. Case () uv Either or u V E E but not both. Then no ege incient at u lies in. (uv) if u V (i.e)if uv E ( )(uv) (uv) if u V (i.e)if uv E So (u) (uv) (u) Hence if u V, then uve an Case () (u) (uv) (u) u V if, then uve u VV but no ege incient at u lies in E E. Then any ege incient at u is either in E or in E but not in E E. : (, ) Also all these eges are inclue in. : (, ) Hence the egree of u in is given by, (u) ( )(uv) uve (uv) (uv) uve uve [ (u) (u)] Case (3) u VV E E E an some eges incient at u are in. By the efinition, any ege in E will not be inclue : (, ) in. Then the egree of u in the irect sum : (, ) is

5 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 5 ISSN (u) ( )(uv) uve (uv) (uv) uve E uve E (uv) (uv) [ (uv) (uv)] [ (uv) (uv)] uve E uve E uve E uve E (uv) (uv) (uv) (uv) [ (uv) (uv)] uv E E uve E uv E E uve E uve E [ (u) (u)] [ (uv) (uv)] uvee From the above two cases we conclue that the egree of the vertex in in terms of the egrees of the vertices in :(, ) an :(, ) is obtaine as follows: (u), if u V V (u), if u V V (u) (u) (u), if u V V an E E [ (u) (u)] [ (uv) (uv)], if u V V an E E uvee Hence the theorem is prove. 4.Example: Consier the two fuzzy graphs :(σ, μ ) an :(σ, μ ) in which the ege sets are isjoint an their sum : (, ). Figure 4: Degree of vertices in the Direct sum of two fuzzy graphs The egrees of the vertices in the irect sum as follows: (u ) (u ) 0.3 (u 3) 0.4 ; ; ; (u 4) (u 5) ; Now let us fin the egrees of the vertices in the irect sum of two fuzzy graphs an in terms of egrees of the vertices in the fuzzy graphs an. E Since there is no ege in E u an V V the egree of u in is exactly the sum of the egrees of u in an.that is, (u ) (u ) (u ) ( ) ( ).5 u The vertices u an u 3 are in V only an not in V. That is,,u3 V V. Hence the egrees of u an u 3 in are equal to the egrees of u an u 3 in. That is, (u ) (u ) 0.3 an (u ) (u ) u,u V V Similarly, since 4 5, we have, (u ) (u ) 0.5 an (u ) (u )

6 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 6 ISSN Example: Here is another example in which the ege sets are not isjoint. Figure 5: Degree of vertices in the Direct sum of two fuzzy graphs Here, we have E E {u u,u u 3 }. From the graph of the irect sum : (, ), we see that the egrees of the vertices of are: (u ) (u 3) (u 4) ; ;. Now we shall fin the egrees of the vertices in in terms of the egrees of the vertices in :(, ) an :(, ). E Since E {uu,uu 3} u the eges u,u u 3 are not in. The vertex u4 V V. Hence by the previous case, the egree of u 4 in is that of u 4 in.that is, (u 4) (u 4) Since E E an u V V (u ) [ (u ) (u )] [ (uu i ) (uu i )] the egree of u is given by, Similarly, since E E an u3 V V the egree of u 3 is given by, (u ) [ (u ) (u )] [ (u u ) (u u )] i 3 i u3uiee [ (u ) (u )] [ (u u ) (u u )] [( ) (0. 0.)] [0.3 0.] uui EE [ (u ) (u )] [ (u u ) (u u )] [( ) ( )] [ ] V. DIRECT SUM OF TWO REULAR FUZZY RAPHS If :(σ, μ ) an :(σ, μ ) are two regular fuzzy graphs then their irect sum : (, ) nee not be a regular fuzzy graph. It is illustrate with the following examples.

7 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 7 ISSN Example: 5.Example: Figure 5: The Direct sum of two regular fuzzy graphs Figure 6: The Direct sum of two regular fuzzy graphs 5.3Remark: : (, ) For the irect sum to be regular, :(σ, μ ) an :(σ, μ ) nee not be regular fuzzy graphs. It is illustrate with the following examples. 5.4Example: 5.5Example: Figure 7: The Direct sum of two non-regular fuzzy graphs 5.6Example: Figure 8: The Direct sum of a regular fuzzy graph an a non-regular fuzzy graph

8 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 8 ISSN Figure 9: The Direct sum of two regular fuzzy graphs From the above examples, we can see that there is no relationship between the regular property of the given fuzzy graphs an the irect sums of them. In the following result, we obtain the necessary an sufficient conition for the irect sum of two regular fuzzy graphs to be regular when V V =. 5.7Theorem: If :(σ, μ ) an :(σ, μ ) are regular fuzzy graphs with egrees k an k respectively an V V = then : (, ) is regular if an only if k =k. Proof: Let :(σ, μ ) be a k -regular fuzzy graph with unerlying crisp graph *:(V, E ) an let :(σ, μ ) be a k -regular fuzzy graph with unerlying crisp graph *:(V, E ) respectively such that V V =. Assume that : (, ) is regular. We know that, (u), if u V V (u), if u V V (u) (u) (u), if u V V an E E [ (u) (u)] [ (uv) (uv)], if u V V an E E uvee Since V V =, (u), if u V (u) (u), if u V k, if u V (u) k, if u V Since : (, ) is regular, k =k. Conversely assume that :(σ, μ ) an :(σ, μ ) are k-regular fuzzy graphs such that V V =. Then the egree of any vertex in the irect sum is given by, (u), if u V (u) (u), if u V k, if u V (u) k, if u V (u) k, u V V Therefore,. Hence : (, ) is regular.

9 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 9 ISSN VI. DIRECT SUM OF CONNECTED FUZZY RAPHS If :(σ, μ ) an :(σ, μ ) are two connecte fuzzy graphs then their irect sum : (, ) nee not be a connecte fuzzy graph. It is illustrate with the following examples. 6.Example: 6.Theorem: If :(σ, μ ) an :(σ, μ ) are two connecte fuzzy graphs with unerlying crisp graphs *:(V, E ) an *:(V, E ) respectively such that E E = an V V then their irect sum : (, ) is a connecte fuzzy graph. Proof: Since :(σ, μ ) is a connecte fuzzy graph, (u,v) 0 for all (u,v) E an since :(σ, μ ) is a connecte fuzzy graph, (u,v) 0 for all (u,v) E. Also V V. Therefore there exists at least one vertex which is in V V. But there is no ege in E E. Hence there exists a path between any two vertices in the irect sum : (, ) of :(σ, μ ) an :(σ, μ ). (u, v) 0 That is for all (u,v) E. This implies that : (, ) is connecte. 6.3Remark: If :(σ, μ ) an :(σ, μ ) are two connecte fuzzy graphs with unerlying crisp graphs *:(V, E ) an *:(V, E ) respectively such that n(v V ) = then their irect sum : (, ) is a connecte fuzzy graph. VII. CONCLUSION In this paper, the irect sum of two fuzzy graphs an is efine. A formula to fin the egree of a vertex in the irect sum : (, ) of two fuzzy graphs :(, ) an :(, ) in terms of the egrees of the vertices in :(, ) an :(, ) is obtaine. This has been illustrate with examples. Also some of the characteristics of the irect sum of effective, regular an connecte fuzzy graphs have been illustrate. Operation on fuzzy graph is a great tool to consier large fuzzy graph as a combination of small fuzzy graphs an to erive its properties from those of the small ones. A step in that irection is mae through this paper. REFERENCES [] Frank Harary, raph Thoery, Narosa / Aison Wesley, Inian Stuent Eition, 988. [] John N. Moeson an Premchan S.Nair, Fuzzy raphs an Fuzzy Hypergraphs, Physica-verlag Heielberg, 000. [3] J.N.Moreson an C.S. Peng, Operations on fuzzy graphs, Information Sciences 79 (994), [4] Nagoorgani. A an Raha. K, Conjunction of Two Fuzzy raphs, International Review of Fuzzy Mathematics, 008, Vol. 3, [5] Nagoorgani. A an Raha. K, Some Properties of Truncations of Fuzzy raphs, Avances in Fuzzy Sets an Systems, 009, Vol.4, No., 5-7. [6] P. Bhattacharya, Some remarks on fuzzy graphs, Pattern Recognition Letter 6 (987), [7] Rosenfel, A. (975) "Fuzzy graphs". In: Zaeh, L.A., Fu, K.S., Tanaka, K., Shimura, M. (es.), Fuzzy Sets an their Applications to Cognitive an Decision Processes, Acaemic Press, New York, ISBN , pp AUTHORS First Author Dr. K. Raha, M.Sc.,M.Phil.,Ph.D., P. & Research Department of Mathematics, Periyar E.V.R. College, Tiruchirapalli rahagac@yahoo.com Secon Author Mr.S. Arumugam, M.Sc.,M.Phil.,B.E.,(Ph.D.), ovt. High School, Thinnanur, Tiruchirapalli anbu.saam@gmail.com & arumugammathematics@gmail.com

10 International Journal of Scientific an Research Publications, Volume 3, Issue 5, May 03 0 ISSN Corresponence Author Dr. K. Raha, M.Sc.,M.Phil.,Ph.D., P. & Research Department of Mathematics, Periyar E.V.R. College, Tiruchirapalli rahagac@yahoo.com & anbu.saam@gmail.com