Average D-distance Between Edges of a Graph

Transcription

1 Indian Journal of Science and Technology, Vol 8(), 5 56, January 05 ISSN (Print) : ISSN (Online) : OI : 07485/ijst/05/v8i/58066 Average -distance Between Edges of a Graph Reddy Babu * and P L N Varma epartment of Mathematics, Koneru Lakshmaiah Education Foundation (KL University), Vaddeswaram, Guntur 5 50, India; reddybabu7@gmailcom epartment of Science & Humanities, Vignan s Foundation for Science, Technology and Research University, Vadlamudi, Guntur 5, India; varma_sh@vignanuniversityorg Abstract The -distance between vertices of a graph G is obtained by considering the path lengths and as well as the degrees of vertices present on the path The average -distance of a connected graph is the average of the -distance between all pairs of vertices of the graph Similarly, the average edge -distance is the average of -distances between all pairs of edges in the graph In this article we study the average edge -distance of a graph We find bounds for average edge -distance which are sharp and also prove some other results Key words: Average -distance, -distance, iameter, 000 Mathematics subject classifications: 05C Introduction The concept of distance is one of the important concepts in study of graphs It is used in isomorphism testing, graph operations, hamiltonicity problems, extremal problems on connectivity ad diameter, convexity in graphs etc istance is the basis of many concepts of symmetry in graphs In addition to the usual distance, d(u,v), between to vertices u,v V(G), we have detour distance, superior distance 5, signal distance 7, degree distance etc In an earlier article 9, the authors introduced the concept of -distance by considering not only path length between vertices, but also the degrees of all vertices present in a path while defining the -distance In a natural way we can extend this concept to -distance between edges also Also we have the concept of average distance in graphs which was introduced by ankelmann 5 In 0, we studied the average distance between vertices of a graph with respect to -distance In this article, we study the average -distance between edges The article is arranged as follows In, we collect some definitions and results for easy reference In, we study some properties of average edge -distance and in 4, we calculated the average -distance between edges for some classes of graphs Preliminaries Throughout this article, by a graph G G(V, E), we mean a non-trivial, finite, undirected graph without multiple edges and loops Unless otherwise specified, all graphs we consider are connected For any unexplained notation and terminology, we refer In this section we given some definitions and state some results for later use We begin with -distance in graphs efinition In a graph G, the degree of a vertex v, deg(v), is the number of edges which are incident with v Similarly we can define the degree of an edge e (u,v) as the number of edges which have a common vertex with the edge e ie deg(e) deg (v) +deg (u) *Author for correspondence

2 Reddy Babu and P L N Varma efinition For any connected graph G, we define Δ(G) max{deg(v) : v V(G)} as the maximum vertex degree of G δ(g) min{deg(v) : v V(G)} as the minimum vertex degree of G Δ (G) max{deg(e) : e E(G)} as the maximum edge degree of G δ (G) min{deg(e) : e E(G)} as the minimum edge degree of G efinition If u,v are vertices of a connected graph G, the -length of a connected u v path s is defined as l ( s) ls ( ) + deg( v) + deg( u) + deg( w) where sum runs over all intermediate vertices w of s and l(s) is the length of the path 4 efinition 4 The -distance, d (u,v) between two vertices u,v of a connected graph G is defined as d (u,v) min{l (s)} where the minimum is taken over all u v paths s inv G In other words, d ( uv, ) min { ls ( ) + deg( u) + deg( v) + deg( w) } where the sum runs over all intermediate vertices w in s and minimum is taken over all u v paths s in G If u,v are two vertices of a graph, then d(u,v) denotes the usual distance between u and v By e e(u,v) in E(G), we mean an edge adjacent with the vertices u and v 5 efinition 5 Let G be a connected graph and let e(u,v ) and f (u,v ) be two edges of G The -distance between these edges is defined as { } ed ( e, f) min d ( u, v ), d ( u, v ), d ( u, v ), d ( u, v ) 6 Remark Observe that ed (e,e ) 0 e,e are neighbor edges ie, they have one common vertex 7 efinition 6 Let G be a connected graph of order n The average distance of G, denoted by μ(g), is defined as m( G) (, ) duv where d(u,v) denotes the { uv, } V distance between the vertices u and v 5 Similarly, we can define the average -distance of a graph as follows: 8 efinition 7 Let G be a connected graph of order n The average -distance between vertices of G, denoted by μ(g), is defined as m ( G) d ( uv, ) where d (u,v) { vu, } V denotes the -distance between the vertices u and v Similarly, we can define 8 the average -distance between edges of a graph as follows: 9 efinition 8 Let G be a connected graph of order n The average - di stance between edges of G, denoted by m ( G ), is defined as G ed e e { e, e} E m ( ) (, ) where ed (e,e ) denotes the -distance between the edges e and e Some more definitions 0 efinition 9 A spanning subgraph is a subgraph of G that contains all the vertices of G efinition 0 Let G be a connected graph of order n having m edges with V(G) {v,v,, v n } The -distance matrix of G, denoted as (G), is defined as ( G) [ d ij, ] n n where dij, d ( vi, vj) is the -distance between the vertices vi and vj Obviously (G) in a n n symmetric matrix with all diagonal entries being zero In a similar manner we can define edge -distance matrix [EM] of G, denoted as ( G ), is defined as ( ) [ ij, ] m m G d where ed, ed ( e, e ) is the edge ij i j -distance between the edges e i and e j and d ii, Thus this is a m m symmetric matrix Further, we have Vol 8 () January 05 wwwindjstorg Indian Journal of Science and Technology 5

3 Average -distance Between Edges of a Graph efinition Let G be a graph, then the average degree of G, denoted as d(g), is given by dg ( ) dv ( ) where d(v) is the V v V degree of the vertex v efinition The total edge -distance [TE] of graph G is the number given by (, ) m m ed ei e j where m is the number i j of edges j i Average Edge -distance In this section we prove some results on average -distance between edges Theorem Let G and G and same diameters If the number of edges in G is more than the number of edges in G then average edge -distance of G is more than average edge -distance of G Proof: Since the diameters of these two graphs are the same, the largest entries in the edge -distance matrix of these graphs are the same The number of the pairs of edges examined is greater in the graph whose edge number is greater And this cause TE value to increase Since these orders are same and the number of edges in G is more than number of edges in G then average edge -distance of G is more than average edge -distance of G ie EG ( > EG ( ) m ( G > m ( G) Theorem Let G and G be two connected graphs of same order and diam (G < diam (G), Then m ( G > m ( G) Proof: Since G,G have same number of vertices and diam(g ) < diam(g ), it is clear that EG ( > EG ( ) Then by theorem, we have m ( G > m ( G) Theorem Let G and G be two connected graphs having same orders and same diameters If m ( G < m ( G) G d G d ( ) < ( ) then Proof: Since d ( G < d ( G) and V(G ) V(G ), we have E(G ) < E(G ) Then by theorem G m G m ( ) < ( ) 4 Theorem 4 Let G and G and diameters If d( G < d( G) then m ( G < m ( G) Proof: d( G < d( G) implies E(G ) < E(G ) then by theorem m ( G < m ( G) 5 Theorem 5 Let G and G and same diameters If the average degree of G is less than average degree of G then m ( G < m ( G) Proof: We have by definition E dv ( ) dg ( ) V As the graphs have same v V order, if d(g ) < d(g ), then E(G ) < E(G ) Hence by theorem, we have m ( G < m ( G) 6 Theorem 6 Let H be a spanning subgraph of G Then m ( H) < m ( G) Proof: Number of the vertices of H will remain the same as the graph itself it is obvious that E(H) < E(G) From theorem, we have m ( H) < m ( G) 4 Results on Some Classes of Graphs Here we calculate the average edge -distance for some classes of graphs 4 Theorem If Pn is the path graph with n( ) vertices and n edges, an( n+ then m ( Pn ) where an is given by the relation nn ( a a + n with a 0 n n Proof: For Pn, the edge -distance matrix, m ( G ) the n n symmetric matrix, is 54 Vol 8 () January 05 wwwindjstorg Indian Journal of Science and Technology

4 Reddy Babu and P L N Varma n n0 n n6 n n0 0 5 n9 n6 n By adding all entries in the upper triangular or lower triangular matrix we get the total edge -distance, which is an(n + in this case, where an n( ) is a constant given by {0,,, 6, 0, 5,,} or recursively an an + n Then an( n+ m ( G) nn ( 4 Theorem For a complete graph Kn with n vertices, the average edge -distance m ( ) is give by ( n )( n )(n m ( Kn) 4 Proof: Every edge taken from Kn has (n ) edge n neighbors and distinct edges (due to end points of this edge) The edge -distance between any edge and its neighbor is zero and the edge -distance between any edge and distinct edge in Kn (n Total n edge -distance TE (n and K n TE ( n )( n )(n m ( Kn) 4 4 Theorem The average edge -distance of complete bipartite graph is mn( n ( m ( m + n + m ( Km, n) ( m+ n)( m+ n Proof: In a complete bipartite graph there are mn edges Any edge taken from complete bipartite graph has (m + n ) edge neighbors and (m (n distinct edges The edge -distance between any edge its neighbor is zero and the edge -distance between any edge and distinct edge in complete bipartite graph mn( m ( n ( m + n + is (m + n + TE mn( n ( m ( m + n + m ( Km, n) Therefore ( m+ n)( m+ n 44 Theorem 4 The edge average -distance of Star graph is m ( st, n) 0 Proof: For star graph one vertex is adjacent to all ot hers So every edge has one common vertex and total edge -distance is zero therefore m ( st, n) 0 Alternatively, we may take m in theorem 4 45 Theorem 5 The edge average -distance of cyclic graph is an m ( Cn) n where an an + 6n 5 with a 5 and ( n )(n+ m ( Cn ( n ) ( n Proof: Case (i) Cyclic graphs of odd order, Cn ( n ) As Cn is regular, the elements of any row in the EM, except the diagonal element, are 0,0,5,5,8,8,,,,5 + ( 4) n then n TE (5 8 ) 5 (n 5) n m ( Cn TE and this can seen to be ( n )(n+ m ( Cn ( n Case (ii): Cyclic graphs of even order C n ( n ) In this case the elements of any row in EM except the diagonal element, are 0,0,5,5,8,8,,,,5 + (n ),5 + (n ) an Like above we can show that m ( Cn) where n an an + 6n 5 with a 5 Vol 8 () January 05 wwwindjstorg Indian Journal of Science and Technology 55

5 Average -distance Between Edges of a Graph 5 References Buckley F, Harary F istance in GraphAddison-Wesley, Longman; 990 Chartrand G, Escuadro H, Zhang P etour distance in graphs J Combin Comput 005; 5:75 94 ankelmann P Average distance and dominating number iscrete Appl Math 997; 80: 5 4 ankelmann P, Mukwembu S, Swart HC Average distance and edge connectivity I* Siam J iscrete Math 008; :9 0 5 oyale JK, Graver JE Mean distance in a graph iscrete Math 977; 7: Kathiresan KM, Marimuthu G Superior distance in graphs J Combin Comput 007; 6: Kathiresan KM, Sumathi R A study on signal distance in graphs Algebra, graph theory and their applications Narosa publishing house Pvt Ltd; 00 p Balci MA, under P Average Edge-istance in Graphs Selcuk Journal of Appl Math 00; : Babu R, Varma PLN -distance in graphs Golden Research Thoughts 0; :5 8 0 Babu R, Varma PLN Average -distance between vertices of a Graph 0 56 Vol 8 () January 05 wwwindjstorg Indian Journal of Science and Technology