The eccentric-distance sum of some graphs

Transcription

1 Electronic Journal of Graph Theory an Applications 5 (1) (017), 51 6 The eccentric-istance sum of some graphs Pamapriya P., Veena Matha Department of Stuies in Mathematics University of Mysore, Manasagangotri Mysuru , Inia pamapriyap7@gmail.com, veena matha@reiffmail.com Abstract Let G = (V, E) be a simple connecte graph. The eccentric-istance sum of G is efine as ξ s (G) = [e(u) + e(v)](u, v), where e(u) is the eccentricity of the vertex u in G {u,v} V (G) an (u, v) is the istance between u an v. In this paper, we establish formulae to calculate the eccentric-istance sum for some graphs, namely wheel, star, broom, lollipop, ouble star, frienship, multi-star graph an the join of P n an P. Keywors: eccentricity, star, path, broom, lollipop graph, ouble star, complete k-partite Mathematics Subject Classification : 05C10 DOI: /ejgta Introuction Let G be a simple connecte graph with the vertex set V (G) an the ege set E(G). The egree of a vertex u V (G) is enote by (u) an is the number of vertices ajacent to u. For vertices u, v V (G), the istance (u, v) is efine as the length of any shortest path connecting u an v in G an D(u) enotes the sum of istances between u an all other vertices of G. The eccentricity e(u) of a vertex u is the largest istance between u an any other vertex v of G, i.e., e(u) = max{(u, v); v V (G)}. Let K n, P n, W n, C n an K 1,n 1 enote a complete graph, path, wheel, cycle an star on n vertices, respectively [14]. Receive: 4 October 016, Revise: 15 January 017, Accepte: 8 January

2 The eccentric-istance sum of some graphs Pamapriya P. et al. The Wiener inex is efine as the sum of all istances between unorere pairs of vertices W (G) = (u, v). {u,v} V (G) It is consiere as one of the most use topological inex with high correlation with many physical an chemical inices of molecular compouns (for the recent survey on Wiener inex see [4, 5]). The parameter DD(G) calle the egree istance of G was introuce by Dobrynin an Kochetova [6] an Gutman [13] as a graph-theoretical escriptor for characterizing alkanes; it can be consiere as a weighte version of the Wiener inex an is efine as DD(G) = [(u) + (v)](u, v) = (u)d(u) {u,v} V (G) u V (G) where the summation goes over all pairs of vertices in G. In fact, when G is a tree on n vertices, it has been emonstrate that Wiener inex an egree istance are closely relate by (see [15, 19]) DD(G) = 4W (G) n(n 1). Sharma, Goswami an Maan [6] introuce a istance-base molecular structure escriptor, eccentric connectivity inex (ECI) efine as ξ c (G) = e(v)(v). v V (G) The inex ξ c (G) was successfully use for mathematical moels of biological activities of iverse nature [9, 11, 1,, 5, 6]. The investigation of its mathematical properties starte only recently (for a survey on eccentric connectivity inex see [17]). In [8, 18, 3, 8], the extremal graphs in various class of graphs with maximal or minimal ECI are etermine. In [1,, 7] the authors etermine the close formulae for the eccentric connectivity inex of nanotubes an nanotori. Recently, a novel graph invariant for preicting biological an physical properties eccentricistance sum was introuce by S.Gupta, M.Singh an A.K.Maan [1]. This topological inex has vast potential application in structure activity/property relationships of molecules an it also isplays high iscriminating power with respect to both biological activities an physical properties; see[1]. The authors in [1] have shown that some structure activity an quantitative structure property stuies using eccentric-istance sum were better than the corresponing values obtaine using the Wiener inex. It is also interesting to stuy the mathematical property of this topological inex. The eccentric-istance sum (EDS) of G is efine as ξ s (G) = u V (G) e(u)d(u). The eccentric-istance sum can be efine alternatively as ξ s (G) = [e(u) + e(v)](u, v). {u,v} V (G) 5

3 The eccentric-istance sum of some graphs Pamapriya P. et al. Yu, Feng an Ilić [7] ientifie the extremal unicyclic graphs of given girth having the minimal an secon minimal EDS; they also characterize trees with minimal EDS among the n-vertex trees of a given iameter. Hua, Xu an Shu[16] obtaine the sharp lower boun on EDS of n-vertex cacti. Ilić, Yu an Feng [0] stuie various lower an upper bouns for the EDS in terms of other graph invariant incluing the Wiener inex, the egree istance inex, the eccentric connectivity inex an so on. In this paper we establish formulae to calculate the eccentric-istance sum for some graphs, namely wheel, star, broom graph, lollipop, ouble star, frienship, multi-star graph an P l n graph.. Main Results M. J. Morgan et al. calculate the eccentric-istance sum for complete graph, cycle graph an path graph. Proposition.1. [3] 1.ξ s (K n ) = n(n 1).ξ s (C n ) = n 4, if n is even 8 n(n 1) (n + 1), if n is o. 8 3.ξ s (P n ) = 5n 4 16n 3 8n + 16n, if n is even 96 5n 4 16n 3 34n + 16n + 9, if n is o. 96 Proposition.. ξ s (W n ) = (n 1)(4n 9) Proof. e(v 1 ) = 1 an e(v i ) = where, i =, 3,..., n. Then, ξ s (W n ) = [e(v i ) + e(v j )](v i, v j ) {v i,v j } V (W n) =(1 + )1(n 1) + ( + )1() + ( + )(n 4) + ( + )1 + ( + )(n 4) + ( + )1 + ( + )(n 5) + + ( + )1 + ( + )(1) + ( + )1 =(n 1)(4n 9). 53

4 The eccentric-istance sum of some graphs Pamapriya P. et al. Proposition.3. ξ s (K 1,n 1 ) = (n 1)(4n 5) Proof. e(v 0 ) = 1 an e(v i ) = where, i = 1,, 3,..., n 1. Then, ξ s (K 1,n 1 ) =(1 + )1(n 1) + ( + )(n ) + ( + )(n 3) + + ( + )(1) =(n 1)(4n 5). Definition.1. [3] The broom graph B n, is a graph consisting of a path P, together with (n ) en vertices all ajacent to the same en vertex of P. u n u 4 v 1 v v 3 v 1 v Figure 1. Broom. graph B n, u 3 u u 1 Theorem.1. The eccentric-istance sum of a broom graph B n, is when is even, when is o. ξ s (P +1 ) + (n ) ( 3 + 4n) + ξ s (P +1 ) + (n ) ( 3 + 4n) + 1 k + ( k)k, k= k= 1 +1 k + ( k)k, Proof. Let {v 1, v,..., v, u 1, u,..., u n } be the set of n vertices of the broom graph B n,. We consier the following cases. Case(i): is even. e(v 1 ) =, e(v ) = e(v ) =, e(v 3 ) = e(v 1 ) =,..., e(v ) = e(v +) = + 1, e(v +1) =, an e(u 1) = e(u ) = = e(u n ) =. Then,

5 The eccentric-istance sum of some graphs Pamapriya P. et al. ξ s (B n, ) =ξ s (P +1 ) + {[ + ] + [() + ] () + [( ) + ] ( ) + [( ) ] ( ) [( ) ] ( ) [ ] [( ) ] ( ) } [( ) + ] + [() + ] 1 (n ) + ( + ) [(n ) + (n ) ] =ξ s (P +1 ) + (n ) 1 ( 3 + 4n) + k + ( k)k. Case(ii): is o. e(v 1 ) =, e(v ) = e(v ) =, e(v 3 ) = e(v 1 ) =,..., e(v +1 k= 1) = e(v +1 +) = +, e(v +1 ) = e(v +1 +1) = + 1, an e(u 1 ) = e(u ) = = e(u n ) =. Then, ξ s (B n, ) =ξ s (P +1 ) + {[ + ] + [() + ] () + [( ) + ] ( ) + [( ) ] ( ) [( ) ] ( ) [( ) ] ( ) [( ) ] ( ) [( ) + ] + [() + ] 1} (n ) + ( + ) [(n ) + (n ) ] =ξ s (P +1 ) + (n ) ( 3 + 4n) + k= 1 +1 k + ( k)k. 1 Definition.. [3] The lollipop graph L n, is a graph obtaine from a complete graph K n an a path P, by joining one of the en vertices of P to all the vertices of K n. Figure. Lollipop. graph L 10,5 55

6 The eccentric-istance sum of some graphs Pamapriya P. et al. Theorem.. The eccentric-istance sum of a lollipop graph L n, is when is even, when is o. ξ s (P +1 ) + (n ) ( + 1) + (n ) + 1 k + ( k)k, ξ s (P +1 ) + (n ) k= ( + 1) + (n ) + k= 1 +1 k + ( k)k, Proof. Let {v 1, v,..., v, u 1, u,..., u n } be the set of n vertices of the lollipop graph L n,. We consier the following cases. Case(i): is even. ξ s (L n, ) =ξ s (P +1 ) + {[ + ] + [() + ] () + [( ) + ] ( ) + [( ) ] ( ) [( ) ] ( ) [ ] [( ) ] ( ) } [( ) + ] + [() + ] 1 (n ) Case(ii): is o. + (n ) (n ) =ξ s (P +1 ) + (n ) 1 ( + 1) + (n ) + 1 k + ( k)k. ξ s (L n, ) =ξ s (P +1 ) + {[ + ] + [() + ] () + [( ) + ] ( ) + [( ) ] ( ) [( ) ] ( ) [( ) ] ( ) [( ) ] ( ) [( ) + ] + [() + ] 1} (n ) + (n ) (n ) 1 =ξ s (P +1 ) + (n ) ( + 1) + (n ) + + ( k)k. k= k= 1 +1 k 56

7 The eccentric-istance sum of some graphs Pamapriya P. et al. Definition.3. [1] A ouble star graph S n,m is a graph constructe from K 1,n 1 an K 1, by joining their centers v 0 an u 0. The vertex-set V (S n,m ) is V (K 1,n 1 ) V (K 1, ) = {v 0, v 1,..., v n 1, u 0, u 1,..., u } an the ege-set E(S n,m ) = {v 0 u 0, v 0 v i, u 0 u j 1 i (n 1); 1 j (m 1)}. Therefore, a ouble star graph is bipartite. v n 1 u v 6 u 6 v 5 u 5 v 4 v u 0 u 4 0 v 3 u 3 v u v 1 u 1 Figure 3. Double star graph S n,m. Theorem.3. The eccentric-istance sum of a ouble star graph S n,m is 3[(n )(n 1) + m(m + 5) + 6(n 1)(m 1) 5]. Proof. e(v 0 ) = = e(u 0 ) an e(v i ) = 3 = e(u j ) where, i = 1,,..., n 1 an j = 1,,..., Then, ξ s (S n,m ) =[(3 + )1 + (3 + ) + (3 + 3)3(m 1)](n 1) + ( + )1 + ( + 3)(m 1) + ( + 3)1(m 1) + (3 + 3)(n ) + (3 + 3)(n 3) + + (3 + 3)(1) + (3 + 3)(m ) + (3 + 3)(m 3) + + (3 + 3)(1) =3[(n )(n 1) + m(m + 5) + 6(n 1)(m 1) 5]. Definition.4. [13] The frienship (or Dutch winmill or fan) graph F n is a graph constructe by joining n copies of the cycle graph C 3 with a common vertex. v 5 v 6 v 4 v 7 v 0 v 3 vn v v1 Figure 4. Frienship. graph F n 57

8 The eccentric-istance sum of some graphs Pamapriya P. et al. Theorem.4. The eccentric-istance sum of frienship graph F n is n(8n 3). Proof. e(v 0 ) = 1 an e(v i ) = where, i = 1,,..., n. Then, ξ s (F n ) =( + )1 + ( + )(n ) + ( + )(n ) + ( + )1 + ( + )(n 4) + ( + )(n 4) + ( + )1 + ( + )(n 6) + ( + )(n 6). + ( + )1 + ( + )() + ( + )() + ( + )1 + (1 + )1(n) =n(8n 3). Definition.5. [4] Consier the star graph K 1,n with vertex set {v 0, v 1, v,..., v n }, introuce an ege to each of the penant vertices v 1, v,..., v n to get the resulting graph K 1,n,n with vertices {v 0, v 1,..., v n, v n+1,..., v n }, again introuce an ege to each of the penant vertices v n+1,..., v n, to get the graph K 1,n,n,n. Repeating this (m 1) times we get a graph K 1,n, n,..., n }{{} m times calle multi-star graph with (mn + 1) vertices v 0, v 1, v,..., v n, v n+1,..., v n, v n+1,..., v 3n,..., v ()n+1,..., v mn an mn eges, as shown in Figure 5. Figure 5. Multi-star graph K. 1,n, n,..., n }{{} m times Theorem.5. The eccentric-istance sum of a multi-star graph K 1,n,n,...,n is { [ ] nm m (m + 1)(8m + 1) + n(n + 1) m (k + 1)k + (m k)(m k) 6 58

9 The eccentric-istance sum of some graphs Pamapriya P. et al. [ m + k + 1 m ] } k(k + 1) + (k + 1)(k + ) n ] + (m + k)(m k)(m k + 1). Proof. ξ s K 1,n, n,..., n }{{} m times [ ={[m + (m + 1)]1 + [m + (m + )] + + [m + (m + m)]m}n (m + k + 1)k(k + 1) + {[(m + 1) + (m + 1)] + [(m + 1) + (m + )]3 + + [(m + 1) + (m + m)](m + 1)} [(n 1) + (n ) + (n 3) ] + {[(m + ) + (m + 1)]3 + [(m + ) + (m + )]4 + + [(m + ) + (m + m)](m + )} [(n 1) + (n ) + (n 3) ].. + {[(m + m) + (m + 1)](m + 1) + [(m + m) + (m + )](m + ) + + [(m + m) + (m + m)](m + m)}[(n 1) + (n ) + (n 3) ] + {[(m + 1) + (m + )]1 + [(m + 1) + (m + 3)] + + [(m + 1) + (m + m)](m 1)}n + {[(m + ) + (m + 3)]1 + [(m + ) + (m + 4)] + + [(m + ) + (m + m)](m )}n + {[(m + (m 1)) + (m + m)]1}n = nm 6 (m + 1)(8m + 1) + n(n + 1) {m [ ] m (k + 1)k + (m k)(m k) [ m + k + 1 m ] } k(k + 1) + (k + 1)(k + ) + 1 [ + n ] (m + k + 1)k(k + 1) + (m + k)(m k)(m k + 1). Definition.6. [4] P l n (n 3) is a graph obtaine by the join of P n an P, as shown in Figure 6. 59

10 The eccentric-istance sum of some graphs Pamapriya P. et al. u 1 u u 3 u 4 u 5 u n 1 u n Figure 6. P l n graph, n 3. Theorem.6. The eccentric-istance sum of P l n (n 6) graph is 4n + 19n 9. Proof. e(u 1 ) = 1 = e(u ) an e(u i ) = where, i = 3, 4,..., n. Then, ξ s (P l n ) =(1 + )1(n ) + (1 + 1)1 + (1 + )1(n ) + ( + )1 + ( + )(n 4) + ( + )1 + ( + )(n 5). + ( + )1 + ( + )(1) + ( + )1 =4n + 19n 9. Acknowlegement The authors are thankful to the University Grants Commission, Government of Inia, for the financial support uner the Basic Science Research Fellowship. UGC vie No.F.5 1/014 15(BSR)/7 349/01(BSR), January 015. References [1] A.R. Ashrafi, T. Došlić an M. Saheli, The eccentric connectivity inex of TUC 4 C 8 (R) nanotubes, MATCH Commun. Math. Comput. Chem. 65 (011), [] A.R. Ashrafi, M. Saheli an M. Ghorbani, The eccentric connectivity inex of nanotubes an nanotori, J. Comput. Appl. Math. 35 (011), [3] M. Azari an A. Iranmanesh, Computing the eccentric-istance sum for graph operations, Discrete Appl. Math. 161 (013), [4] A. Dobrynin, R. Entringer an I. Gutman, Wiener inex of trees: theory an applications, Acta Appl. Math. 66 (001),

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