Medha Itagi Huilgol a, M. Rajeshwari b & S. Syed Asif Ulla a a Department of Mathematics, Bangalore University, Central

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3 Products of distance degree regular and distance degree inective graphs Medha Itagi Huilgol, * M. Raeshwari, S. Syed Asif Ulla, Departent of Matheatics Bangalore University Central College Capus Bangalore India Departent of Matheatics Acharya Institute of Technology Bangalore India Abstract The eccentricity eu ( ) of a vertex u is the axiu distance of u to any other vertex in G. The distance degree sequence (dds) of a vertex v in a graph G = ( V, E) is a list of the nuber of vertices at distance,,.., eu ( ) in that order, where eu ( ) denotes the eccentricity of u in G. Thus the sequence ^di0, di, dif, di, fh is the dds of the vertex v i in G where d i denotes nuber of vertices at distance fro v i. A graph is distance degree regular (DDR) graph if all vertices have the sae dds. A graph is distance degree inective (DDI) graph if no two vertices have sae dds. In this paper we consider Cartesian and noral products of DDR and DDI graphs. Soe structural results have been obtained along with soe characterizations. Keywords: Distance degree sequence, Distance degree regular (DDR) graphs, Distance degree inective (DDI) graphs, Cartesian and Noral product of graphs. * E-ail: edha@bub.ernet.in E-ail: raeshwari@acharya.ac.in E-ail: syedasif.ulla84@gail.co Journal of Discrete Matheatical Sciences & Cryptography Vol. 5 (0), No. 4 & 5, pp Taru Publications

4 304 MEDHA. I. HUILGOL, M. RAJESHWARI AND S. SYED ASIF ULLA. Introduction Unless entioned otherwise for terinology and notation the reader ay refer Buckley and Harary [4], new ones will be introduced as and when found necessary. Products in graphs have always given ore generalized results copared to the graphs involved in the product itself. It is also a powerful tool to construct bigger graphs given saller ordered (sized) graphs. Many paraeters are tested in the products in literature [6], [7], [8], etc. Aong any products defined between graphs the cartesian product is the ost used one. Recently a whole onograph by Irich et. al., [0] is dedicated to graphs and their cartesian product. The cartesian product is defined as, The cartesian product of two graphs G and H, denoted GX H, is a graph with vertex set VG ( X H) = VG ( )# VH ( ), that is, the set g! V( G)& h! V( H). The edge set of GX H consists of all pairs [( g, h),( g, h )] of vertices with [ g, g ]! E( G) and h = h or g = g and [ h, h ]! E( H). Also the noral product is defined as, The noral product of two graphs G and H, denoted G5 H, is a graph with vertex set VG ( 5 H) = VG ( )# VH ( ), that is, the set g! V( G), h! V( H) and an edge [( g, h), ( g, h )] exists whenever any of the following conditions hold good : (i) [ g, g ]! E( G)and h = h, (ii) g = g and [ h, h ]! E( H), (iii) [ g, g]! E( G) & [ h, h]! E( H). The distance duv (,) fro a vertex u of G to a vertex v is the length of a shortest u to v path. The eccentricity ev () of v is the distance to a farthest vertex fro v. If, dist(,) u v = e(),( u v! u), we say that v is an eccentric vertex of u. The radius rg ( ) is the iniu eccentricity of the vertices, whereas the diaeter dg ( ) is the axiu eccentricity. A vertex v is a central vertex if ev () = rg ( ), and a vertex is an antipodal vertex if ev () = dg ( ). A graph is self-centered if every vertex has the sae eccentricity, i.e., rg ( ) = dg ( ). The distance degree sequence ( dds ) of a vertex v in a graph G = ( V, E) is a list of the nuber of vertices at distance,,., ev ( ) in that order, where ev () denotes the eccentricity of v in G. Thus, the sequence ( di0, di, di.., di,.) is the dds of the vertex v i in G where, di denotes

5 PRODUCTS OF DISTANCE DEGREE GRAPHS 305 nuber of vertices at distance fro v i. The concept of distance degree regular (DDR) graphs was introduced by G. S. Bloo et. al., [], as the graphs for which all vertices have the sae dds. For exaple, the three diensional cube Q3 = K# K# K is a DDR graph with each vertex having its dds as (,3,3,). Clearly, di denotes the degree of the vertex v i in G and hence, in general, a DDR graph ust be a regular graph; but, it is easy to verify that a regular graph ay not be DDR. In Bloo [], detailed study of DDR graphs can be found and one of the fundaental results therein states that Every regular graph with diaeter at ost two is DDR. Bloo, Quintas and Kennedy [] have dealt any probles concerning distance and path degree sequences in graphs. Halbersta et. al., [9] have dealt in particular the distance and path degree sequences for cubic graphs. It is worth to ention that coputer investigation and generation of cubic graphs is done by Brinkann [3] and Busseaker et. al., [5]. In [], Itagi Huilgol et. al., have listed all DDR graphs of diaeter three with extreal degree regularity. In the sae paper they have shown the existence of a diaeter three DDR graph of any arbitrary degree regularity. In [3], Itagi Huilgol et. al., have constructed DDR graphs of arbitrary diaeter. Also, they have studied the DDR graphs with respect to other paraeters. A graph is distance degree inective (DDI) graph if no two vertices have sae dds. These graphs were defined by G.S. Bloo et. al., in []. DDI graphs being highly irregular, in coparison with the DDR graphs, at least the degree regularity is looked into by Jiri volf in []. A particular case of cubic DDI graphs is considered by Martenez and Quintas in [4]. There are very few exaples of DDI graphs, so it is iportant to get DDI graphs fro saller (sized/ordered) DDI or other graphs as products. In this paper we consider cartesian and noral products of DDR and DDI graphs. For soe products both necessary and sufficient conditions have been obtained.. Cartesian product of DDR and DDI graph In this section we consider cartesian products of DDR and DDI graphs. Theore.: Cartesian product of two graphs G and G is a DDR graph if and only if both G and G are DDR graphs.

6 306 MEDHA. I. HUILGOL, M. RAJESHWARI AND S. SYED ASIF ULLA Proof. Let G and G be two DDR graphs having the dds of each vertex ( d0, d, d, dr ) and ( d0l, dl, dl, dl r ) respectively, where r and r are radii of G and G respectively. In the cartesian product of any two graphs, the distance between any two vertices ( u, v ) and ( u, v ) is given by dgx G (( u, v), ( u, v)) = dg ( u, u) + dg( v, v) as in [5]. Now let u be any vertex in G and v be any vertex in G. Then, it is very clear that the nuber of vertices at distance i fro ( uv,) in G 4G i - d ( u, v) = d ( u) + dl( v) + d ( u) dl ( v). / ig XG i i i- = Since the graphs G and G are DDR graphs di() u = di(), x 0 # i # dia( G ) for all x! G and dli() v = dl i(), y 0 # i # dia( G) for all y! G, di ( u, v) = di (,), s t 0 # i # dia( G) + dia( G) G 4 G G 4 for all ( st,)! G GXG. Hence the graph GXG is a DDR graph. Now let GXG, the cartesian product of G and G be a DDR graph. Suppose G is not a DDR graph, then there exist at least two vertices u and v having different dds i.e., ( d0( u), d( u), d ( u), deu ( )( u)) and ( d0( v), d( v), d ( v), dev ()( v)) are dds of u and v, respectively in G and k, the iniu value of i, # i # d( G), such that dk() u! dk() v. Let w be any vertex in G, having the dds ( dl0( w), dl ( w), dl( w),, dl ew ( )( w)) and dkl ( w) be the nuber of vertices at distance k fro w in G. The nuber of vertices at distance k fro (, uw ) in GX G is given by dkg (, u w) = X G d ( ) + dl( w) + d ( u) dl ( w) + d ( u) dl ( w) + d () u d 3 ( w) + f + k u k ( k-) ( k-) 3 l( k - ) d( k - ) () u dl ( w) and the nuber of vertices at distance k fro (, vw ) in GX G is given by dk (, v w) = dk() v + dl( w) + d ( v) d GX G k l ( k- ) ( w ) + d () v dl ( w) + d3() v dl 3 ( w) + f + d ( v) dl ( w). Hence dk ( k - ) ( k- ) ( k-) GX G (, uw )! dkg (, v w), X G since d () u d () v k! k and d() u = d(), v for all,0 # # k. Hence GX G is non-ddr graph, a contradiction. Hence G should be a DDR graph. Siilarly we can prove G is also a DDR graph. Hence, the result. 4 Theore.: If the cartesian product of two graphs G and G is DDI then both G and G are DDI graphs. Proof. Let GX G be a DDI graph. Suppose G is not a DDI graph, then there exist at least two vertices u, u in G having the sae dds, i.e., dds( u) = dds( u ). Let v be any vertex in G. Then the nuber of vertices at distance l,0 # l# eu ( ) + ev ( ) fro ( u, v ) is given by

7 PRODUCTS OF DISTANCE DEGREE GRAPHS 307 i - d ( u, v ) = d ( u ) + dl X ( v ) + / d ( u ) dl ( v ). lg G i i i- = and the nuber of vertices at distance l,0 # l # e( u) + e( v)fro( u, v) is given by i - d ( u, v ) = d ( u ) + dl( v ) + d ( u ) dl X ( v ). lg G i i i- = / Since, dds( u) = dds( u ), we get dlg G( u, v) = dlg G( u, v), X X for all l,0 # l # e( u) + e( v), i. e., dds( u, v) = dds( u, v ), hence GX G is not DDI, a contradiction. Hence G should be DDI. Siilarly, we can prove G is also DDI. Hence, the proof. 4 Reark. Cartesian product of two DDI graphs need not be DDI. The following are the two DDI graphs whose cartesian product is not DDI. Figure Two DDI graphs whose cartesian product is not DDI Lea.: Let G and G be two DDI graphs. Let A = " dds( ui)/ ui! V( G), and B " dds( v )/ v! V( G ),. If A+ B $ then GX G is not DDI. = i i Proof. Let A+ B $ Then there exist uk, ul in G and v, vn in G such that dds( uk) = dds( v) and dds( ul) = dds( vn). Hence in GX G, dds( uk, vn ) = dds( ul, v), aking GX G non DDI. Hence, the proof 4 Theore.3. Let G and G be any two graphs. Let u be any vertex in G and S be a subset of VG ( ) such that no two vertices of S have sae dds, then no two vertices of { u }# S have sae dds in GX G. Proof. Let u be any vertex in G and S be a subset of VG ( ) such that no two vertices of S have sae dds. Suppose there exist at least two vertices ( uv,) and ( uw, ) in { u }# S having sae dds. Hence dlg (, u v) = dl (, u w), 4G G 4G for all l,0 # l# eu ( ) + ev ( ), here ev () = ew ( ).

8 308 MEDHA. I. HUILGOL, M. RAJESHWARI AND S. SYED ASIF ULLA l - Hence dl() u + dl() v + / d() u dl () v = dl() u + dl( w) + / d () u d ( w), l l - = l- l l - = iplies dl () v + / d() u dl - () v = dl ( w) + / d( u) dl - ( w) " ( ) = For l =, eq.() iplies dl() v = dl ( w). l l l - = l - For l =, eq.() iplies dl() v + d() u dl () v = dl( w) + d() u dl ( w) & dl () v = dl ( w) and so on For, l = e() v = e( w), eq.() iplies dlev () () v = dl e( w) ( w). Hence, dds() v = dds( w), a contradiction. Hence no two vertices of { u }# S have sae dds in G4G. Hence, the proof. 4 Reark. Let S and S be two subsets of VG ( ) such that every pair (,), xy x! S, y! S satisfies dds() x! dds() y and let z! V( G ) be any vertex in G, then in G4G, the subsets {( xz, ) / x! S } and "(,)/ yz y! S, are such that every pair (( xz, ),( yz, )), x! S and y! S satisfies dds(,) x z! dds(,) y z. 3. Noral product of DDR and DDI graphs In this section we consider noral product of DDR and DDI graphs. Stevanović in [6] has considered the distance between any pair of vertices in noral product. Given two vertices ( u i, v ) and ( u k, v ) the distance between these two vertices in the noral product is given by; d (( u, v ),( u, v ) { d ( u, v ), d ( u, v )} G5 G i k = ax G i k G Iediate conclusion we can draw as follows: Lea 3.: Let S = " Gi / i $,. If there exist k such that Gk is self centred and dia( Gk) $ dia( Gi) for all i $ then noral product of all the graphs in S is a self centred graph with diaeter equal to dia( Gk ). Theore 3.: Noral product G 5 G of two graphs G and G is DDR if and only if both G and G are DDR graphs.

9 PRODUCTS OF DISTANCE DEGREE GRAPHS 309 Proof. Let G and G be two DDR graphs having the dds ( d0, d, d, d dg ( )) and ( dl0, dl, dl,, dd( G ) ) respectively, then the nuber of vertices at distance i,0 # i # ax{ dia( G ), dia( G)} fro any vertex u, v in G 5 G is given by l i - i - ig 5 G l = i l : l i + i l / l + li / l d ( u, v ) d ( u) d ( v ) d ( u) d ( v ) d ( v ) d ( u). Hence, the Noral product G 5 G is DDR. Conversely, let G 5 G be DDR. Suppose G is not DDR, then there exist at least two vertices u and u in G such that dds( u)! dds( u ). Let k be the iniu value such that dk( u)! dk( u) and v be any arbitrary vertex in G, then the nuber of vertices at distance k fro ( u, v ) is given by i - k - kg 5 G = k : l k + k / l + lk / d ( u, v ) d ( u ) d ( v ) d ( u ) d ( v ) d ( v ) d ( u ) and the nuber of vertices at distance k fro ( u, v ) is given by i - k - / / d ( u, v ) = d ( u ) dl( v ) + d ( u ) dl( v ) + dl ( v ) d ( u ), kg 5 G k k k k iplies dkg 5G( u, v)! dkg 5G( u, v), since dk( u)! dk( u). So G 5 G is not DDR, a contradiction. Hence G is DDR. Siilarly we can prove G is also DDR. Hence, the proof. 4 Proposition 3.: If the noral product G 5 G of two graphs is DDI then both G and G are DDI. Proof. Let the noral product G 5 G of two graphs be DDI. Suppose G is not DDI, then there exist two vertices u and u such that dds( u) = dds( u ). Let v be any vertex in G. The nuber of vertices at distance k, 0 # k# ax { eu ( ), ev ( )} fro ( u, v ) is given by k - k - / / d ( u, v ) = d ( u ) dl( v ) + d ( u ) dl( v ) + dl( v ) d ( u ). kg+ G k k k k = 0 = 0

10 30 MEDHA. I. HUILGOL, M. RAJESHWARI AND S. SYED ASIF ULLA and the nuber of vertices at distance k, 0 # k# ax{ eu ( ), ev ( )} fro ( u, v ) is given by k - k - / / d ( u, v ) = d ( u ) dl( v ) + d ( u ) dl( v ) + dl ( v ) d ( u ). kg 5 G k k k k = 0 = 0 Hence, dkg 5G( u, v) = dk G 5G( u, v), 0 # k # ax { e( u) = eu ( ), ev ( )}, iplies dds( u, v) = dds( u, v ), a contradiction. Hence G is DDI. Siilarly we can prove G is also DDI. Hence, the proof. 4 Reark 3 : Noral product of two DDI graphs need not be DDI. The graphs in Figure are the two DDI graphs whose noral product is not DDI. Lea 3.: Let G and G be two DDI graphs. Let A = " dds( ui )/ ui! G, and B " dds( vi)/ vi! V( G ),. If A+ B $ then G 5 G is not DDI. = Proof. Let A+ B $. Then there exist uk, ul in G and v, vn in G such that dds( uk ) = dds( v) and dds( ul ) = dds( vn). Hence in G 5 G, dds( uk, vn) = dds( ul, v ), aking G 5 G non DDI. Hence the proof. 4 Proposition 3.: Let G and G be any two graphs. Let u be any vertex in G and S be a subset of VG ( ) such that no two vertices of S have sae dds, then no two vertices of " u, # S have sae dds in G 5 G. Proof. Suppose there exist two vertices ( uv,) and ( uw, ) in " u, # S having sae dds, i.e., dds(,) u v = dds(, u w), this condition is satisfied only if ev () = ew ( ). Here three subcases arise, viz. Case(a): eu () ev () = ew ( ), Case(b): eu () = ev () = ew ( ) and Case(c): eu () ev () = ew ( ). Case(a): dds( u, v) = dds( u, w), e( u) e( v) = e( w), e( u, v) = e( v) = e( w). - dg 5 G (, u v) = d () u dl () v + d () u dl() v + dl () v d () u = 0 - / / and () = 0 - d (, u w) = d () u dl ( w) + d () u dl( w) + dl ( w) d (), u G 5 G - / / () dg 5G (,) u v = dg 5G (, u w), for all, 0 # # ev ( ) = ew ( ).

11 PRODUCTS OF DISTANCE DEGREE GRAPHS 3 First taking, dg 5G (, u v) = dg 5G (, u w), for all,0 # # e( u), we - get d()[ u dl () v - dl ( w)] + d() u = dl() v - dl( w) G - / + = d ()[ u dl() v - dl( w)] G = 0 - / / l () l () l l / / / i.e., = d( u) + d ( u) G 6 d v - d ( + d u = d ( v) - d ( w) G = 0 - i.e., = d ( u) G 6 dl ( v) - dl ( + d ( u) = dl( v) - dl( w) G = 0 - / / / (3) Put = in Eqn (3). ( 6d () u + d () 6dl() v -dl( i.e., dl() v = dl( w) 0 + d () u 6 dl() v - dl( = 0 / 0 0 Put = in Eqn. (3). ( = d () u G 6dl() v - dl( + d() u i.e., dl() v = dl ( w), And so on, = 0 = 0 = dl() v - dl( w) G = 0 / / eu ( ) Put = e() u in Eqn. (). ( = d () u G 6dl () v - dl ( / eu ( ) eu ( ) i.e., dl () v dl ( w). eu ( ) = eu ( ) eu ( )- eu ( )- / / + deu ( )() u = dl() v - dl( w) G = 0 Hence dl () v = dl ( w) for all, 0 # # e( u). (4) Now for all eu, ( ) # ev ( ) = euw (, ), we have

12 3 MEDHA. I. HUILGOL, M. RAJESHWARI AND S. SYED ASIF ULLA eu ( ) d (, u v) = d () v d () u eu ( ) d (, u w) = dl ( w) / d (). u G 5 G l / and G 5 G Hence dg 5G (, u v) = dg 5G (, u w), for all eu, ( ) # ev ( ) eu ( ) eu ( ) / /, i.e.., = euw (, ) gives dl () v d() u - dl ( w) d() u = 0 eu ( ) 6 dl () v - dl ( / d () u = 0 i.e., dl () v - dl ( w) = 0. Hence dl () v = dl ( w), for all eu, ( ) # ev ( ) = euw (, ). (5) Cobining (4) and (5), we get dl() v = dl ( w), for all,0 # # e( v) = ew ( ) = euw (, ), i.e., dds() v = dds( w), a contradiction. Hence dds(,) u v! dds(, u w). Case (b): dds( u, v) = dds( u, w), e( u) = e( v) = e( w) = e( u, v). Substituting these conditions in () and (), we get = d () u G 6 dl () v - dl ( + d () u = dl() v - dl( w) G = / / / (6) Substituting values of in Eq.(6) we get dl() v = dl ( w) for all,0 # # eu () = ev () = ew ( ) = euv (,). i.e., dds() v = dds( w), a contradiction. Hence dds(,) u v! dds(, u w). Case (c): eu () ev () = ew ( ), euv (,) = eu (). Substituting the values in () and () we get - - / / / (7) = d () u G 6 dl () v - dl ( + d () u = dl() v - dl( w) G = 0

13 PRODUCTS OF DISTANCE DEGREE GRAPHS 33 Substituting the values of in (7) we get dl () v = dl ( w) for all, 0 # # ev ( ) = ew ( ). i.e., dds() v = dds( w), a contradiction. Hence, dds(,) u v! dds(, u w). 4 Reark 4: Let S and S be two subsets of V^Gh such that every pair (,), xy x! S, y! S satisfies dds() x! dds() y and let z e V^Gh be any vertex in G, then in G5 G, the subsets "(,)/ xz xe S, and "(,)/ yz ye S, are such that every pair (( xz, ), ( yz, )), xe S and ye S satisfies dds(,) x z! dds(,) y z. 4. Acknowledgeent The third author (SAUS) thanks the UGC, New Delhi for granting fellowship under Special Assistance Progra CAS Phase-II. References [] G. S. Bloo, L. V. Quintas and J. W. Kennedy, Distance Degree Regular Graphs, The Theory and Applications of Graphs, 4th International conference, Western Michigan University, Kalaazoo, MI, May (980), John Wiley and Sons, New York, 98, pp [] G. S. Bloo, L. V. Quintas and J. W. Kennedy, Soe probles concerning distance and path degree sequences, Lecture Notes in Math. 08, 983, pp [3] G. Brinkann, Fast generation of Cubic graphs, J. Graph Theory, Vol. 3(), 996, pp [4] F. Buckley and F. Harary, Distance in Graphs, Addison Wesley 990. [5] F. C. Busseaker, S. Cobelic, D. B. Cvetkovic, J. J. Seidel, Coputer investigation of cube Graphs, T.H. Report 76-WSK-0, Technological University Eindhoven, Netherlands, 976. [6] T. Feder, Stable Network and Product Graphs, Me. Aer. Math. Soc., 6, pp. xii , [7] R. L. Graha and P. M. Winkler, On isoorphic ebeddings of graphs, Trans. Aer. Math. Soc., Vol. 88, 985, pp [8] A. Graovac and T. Pisanski, On the wiener index of a graph, J. Math. Che, Vol. 8, 99, pp

14 34 MEDHA. I. HUILGOL, M. RAJESHWARI AND S. SYED ASIF ULLA [9] F. Y. Halbersta and L. V. Quintas, Distance and path degree sequences for cubic graphs, Matheatics Departent, Pace University, New York, NY, 98. [0] W. Irich, S. Klavžar, D. F. Rall, Topics in Graph Theory: Graphs and Their Cartesian Product, A K Peters, Ltd, 008. [] Jiri volf, A sall distance degree inective cubic graphs, Notes fro New York Graph Theory Day XIII, New York Acadey of Sciences, 987, pp [] Medha Itagi Huilgol, H. B. Walikar, B. D. Acharya, On diaeter three distance degree regular graphs, Advances and Applications in Discrete Matheatics, Vol. 7(), 0, pp [3] Medha Itagi Huilgol, Raeshwari M. and Syed Asif Ulla S., Distance degree regular graphs and their eccentric digraphs, International J. of Math. Sci. and Engg. Appls.(IJMSEA), Vol. 5(VI), 0, pp [4] P. Martinez and L. V. Quintas, Distance degree inective regular graphs, Notes fro New York Graph Theory Day VIII, New York Acadey of Sciences, 984, pp. 9. [5] J. Nieinen, The center proble in the product of graphs, Recent Studies in Graph Theory, 989, pp [6] D. Stevanović, Distance regularity of copositions of graphs, Applied Matheatics Letters, Vol. 7(3), March 004, pp Received Deceber, 0