Some results on the reciprocal sum-degree distance of graphs

Transcription

1 J Comb Optim (015) 30: DOI /s Some results on the reciprocal sum-degree distance of graphs Guifu Su Liming Xiong Xiaofeng Su Xianglian Chen Published online: 17 July 013 The Author(s) 013. This article is published with open access at Springerlink.com Abstract In this contribution, we first investigate sharp bounds for the reciprocal sum-degree distance of graphs with a given matching number. The corresponding extremal graphs are characterized completely. Then we explore the k-decomposition for the reciprocal sum-degree distance. Finally, we establish formulas for the reciprocal sum-degree distance of join and the Cartesian product of graphs. Keywords The reciprocal sum-degree distance Harary index Matching number k-decomposition Join graphs Cartesian product graphs 1 Introduction Throughout the paper, we consider finite undirected simple connected graphs. Let G = (V, E) be a graph, we denote simply the order and size with V and E if no G. Su (B) L. Xiong School of Mathematics, Beijing Institute of Technology, Beijing , People s Republic of China gfs1983@16.com L. Xiong lmxiong@bit.edu.cn G. Su Computational Systems Biology Lab, Department of Biochemistry and Molecular Biology, University of Georgia, Athens, GA 3060, USA X. Su College of Arts and Science, Shanghai Maritime University, Shanghai 01306, People s Republic of China umpire.su@gmail.com X. Chen Department of Mathematics, Changji University, Xinjiang , People s Republic of China xjxianglian@163.com 13

2 436 J Comb Optim (015) 30: Table 1 A family of distanceand degree-based graph invariants W DD + DD H RDD +? RDD? ambiguity can arise. The degree of a vertex u V is the number of edges incident to u, denoted by deg G (u). The maximum and minimum vertex degree in the graph G will be denoted by (G) and δ(g), respectively. The distance between two vertices u and v is the length of a shortest path connecting them in G, denoted by dist G (u,v). The maximum value of such numbers, diam(g), is said to be the diameter of G. A matching of a graph G is a set of edges with no shared endpoints. A maximal matching in a graph is a matching whose cardinality cannot be increased by adding an edge. The matching number β(g) is the number of edges in a maximum matching. The link, denoted by (G 1 G )(a b), of two disjoint graphs G 1 and G is the graph obtained by joining a V (G 1 ) and b V (G ) by an edge. Other terminology and notations needed will be introduced as it naturally occurs in the following and we use (Bondy and Murty 1976) for those not defined here. The motivation for studying the quantity that the authors intend to call reciprocal (sum)-degree distance and reciprocal product-degree distance of a graph respectively, comes from the following observation. The sum of distances between all pairs of vertices in a graph G is the Wiener index (Wiener 1947a), namely W = W (G) = {u,v} V (G) dist G (u,v), (1) was first time introduced by Wiener more that 60 years ago (Wiener 1947a). Initially, the Wiener index W (G) was considered as a molecular-structure descriptor used in chemical applications, but soon it attracted the interest of pure mathematicians (Entringer et al. 1976); for details and additional references see the review (Dobrynin et al. 001) and the recent papers (Caporossi et al. 01; Li et al. 011; Wiener 1947b, c). Eventually, a number of modifications of the Wiener index were proposed, which we present in the following table: In Table 1, W is the ordinary Wiener index, Eq. (1), whereas DD + = DD + (G) = DD = DD (G) = H = H(G) = {u,v} V (G) {u,v} V (G) {u,v} V (G) u =v [ degg (u) + deg G (v) ] dist G (u,v) () [ degg (u) deg G (v) ] dist G (u,v) (3) 1 dist G (u,v) The graph invariants defined via Eqs. () (4) have all been much studied in the past. The invariant DD + was first time introduced by Dobrynin and Kochetova (1994) and named (sum)-degree distance. Later the same quantity was examined under the name 13 (4)

3 J Comb Optim (015) 30: Schulz index (Gutman 1994). For mathematical research on degree distance see (Ilić et al. 011; Tomescu 010) and the references cited therein. A remarkable property of DD + is that in the case of trees of order n, the identity DD + = 4W n(n 1) holds (Klein et al. 199). In Gutman (1994) it was shown that also the multiplicative variant of the degree distance, namely DD,Eq.(3), obeys an analogous relation: DD = 4W (n 1)(n 1). This latter quantity is sometimes referred to as the Gutman index (see Feng and Liu 011; Mukwembi 01; and the references quoted therein), but here we call it product-degree distance. The greatest contributions to the Wiener index, Eq. (1), come from most distant vertex pairs. Because in many applications of graph invariants it is preferred that the contribution of vertex pairs diminishes with distance, the Wiener index was modified according to Eq. (4). This distance-based graph invariant is called Harary index and was introduced in the 1990s by Plavšić etal.(1993). It also was subject of numerous mathematical studies (see Cui and Liu 01; Xu 01; Xu and Das 011; Zhou et al. 008; and the references cited therein). The graph invariants, defined via Eqs. (1) (4), can be arranged as in Table 1.From this Table it is immediately seen that one more such invariants are missing. In Su et al. (01a) introduced the reciprocal product-degree distance of graphs, which can be seen as a product-degree-weight version of Harary index: RDD = RDD (G) = {u,v} V (G) u =v deg G (u) deg G (v). (5) dist G (u,v) They mainly determined the connected graph of given order with maximum RDD - value, and established various lower and upper bounds for RDD in terms of matching, independence number, vertex-connectivity and other topological invariants. Hua and Zhang (01) proposed a new graph invariant named reciprocal degree distance (here we call it reciprocal sum-degree distance), which can be seen as a (sum)-degree-weight version of Harary index: RDD + = RDD + (G) = {u,v} V (G) u =v deg G (u) + deg G (v) dist G (u,v). (6) In (Hua and Zhang 01) they mainly presented some extremal properties of reciprocal degree distance. However, to our best knowledge, the RDD + -value of connected graphs with a given matching number has not been considered by other authors so far. In this contribution, we first investigate sharp bounds of the reciprocal sum-degree distance of graphs with a given matching number. The extremal graphs also determined completely. Then we explore the k-decomposition of the complete graph K n for the reciprocal sum-degree distance. Formulas for the reciprocal sum-degree distance of join and the Cartesian product graphs were also established. 13

4 438 J Comb Optim (015) 30: RDD + -value with given matching number Let G e denote the graph formed from G by deleting an edge e E, and G + e denote the one obtained from G by adding an edge e E. Lemma.1 (Hua and Zhang 01) Let G be a connected graph of order n at least three. Each of the following holds: (a) if G is not isomorphic to K n (the complete graph of order n), then RDD + (G) < RDD + (G + e) for any e E(G); (b) if G has an edge e not being a bridge, then RDD + (G) >RDD + (G e). A component of a graph is said to be odd (resp., even) if it has odd (resp., even) number of vertices. Indicate the number of odd components by o(g). The following is an immediate consequence of the Tutte-Berge formula (Lovász and Plummer 1986). Lemma. (Lovász and Plummer 1986) Let G be a connected graph of order n. Then n β = max{o(g X) X :X V }. Let Q(n,β)denote the class of connected graphs of order n with matching number β. Lemma.3 Let G be a connected graph of order n 4 with matching number β [, n ]. Let σ = 3 4n + 37n 11n Each of the following holds: (a) if β = n, then RDD +(G) n(n 1), with equality if and only if G = K n ; (b) if σ<β n 1, then RDD +(G) 4β 3 +(n 1)β +(11 3n)β+ n n 4, with equality if and only if G = K 1 + (K β 1 K n β ); (c) if β = σ, then RDD + (G) 4σ 3 + (n 1)σ + (11 3n)σ + n n 4 = 1 σ 3 1 σ + n 3n+ σ, with equality if and only if G = K β + K n β or G = K 1 + (K β 1 K n β ); (d) if β<σ,then RDD + (G) 1 β3 1 β + n 3n+ β, with equality if and only if G = K β + K n β. Proof Let G be a connected graph with maximum RDD + -value in Q(n,β).From Lemma., it follows that there exists a vertex subset X V (G ) such that n β = max{o(g X) X :X V }=o(g X ) X. For simplicity, let X =s and o(g X ) = t. Then n β = t s. 13

5 J Comb Optim (015) 30: Case I. s = 0. In this case, G X = G and then n β = t 1. By our choice of G and Lemma.1, we obtain that G = K n, then we have RDD + (G ) = n(n 1). Case II. s 1. Consequently, t 1. Let G o 1, Go,, Go t be all the odd components of G X. To obtain our result, we state and prove the following four claims. Claim 1. There is no even component in G X. To the contrary, let G e be an even component. Then the link (G e Gi o )(a b), obtained by joining a vertex a V (G e ) and b V (Gi o ), is also an odd component in G X. We denote such a graph by G, for which n β(g ) o(g X ) X = o(g X ) X holds. This implies that G Q(n,β), which contradicts to the choice of G. Claim. Each odd component Gi o(1 i t), and the graph induced by X, are complete. Assume that Gi o is not complete. Then there must exist two non-adjacent vertices u,vin Gi o. By Lemma.1, one can get a graph G + uv, which increases the RDD + - value. This again is a contradiction with the choice of G. Similarly, we can prove that X is complete. Claim 3. Each vertex of Gi o is adjacent to a vertex of X. The proof follows as before. Now, without loss of generality, we let n i = V (Gi o ) for i = 1,,, t. Then by Claim 1, and 3 we have G = K s + (K n1 K n K nt ). Let RDD + (G 1, G ) denote the contribution to the RDD + -value between vertices of G 1 and those of G, then we have RDD + (K ni, K ni ) = ( n i ) (ni + s 1), RDD + (K ni, K n j ) = n i n j (n i + s 1 + n j + s 1), RDD + (K ni, K s ) = sn i (n 1 + n i + s 1), RDD + (K s, K s ) = ( s ) (n 1). Hence, the reciprocal sum-degree distance of G can be represented as t RDD + (G ) = RDD + (K ni, K ni ) + RDD + (K ni, K n j ) i< j t + RDD + (K ni, K s ) + RDD + (K s, K s ) 13

6 440 J Comb Optim (015) 30: t t = ni 3 + (s ) ni + (s + (n 3)s + 1) + ( ) s n i n j (n i + n j + s ) + (n 1). i< j t n i We also need claim 4 below, which can be verified by Lagrange multiplier. Claim 4. Each of the following function F 1 (n 1, n,, n t ) = n n3 + +n3 t F (n 1, n,, n t ) = n 1 + n + +n t F 3 (n 1, n,, n t ) = n 1 + n + +n t F 4 (n 1, n,, n t ) = i< j n in j (n i + n j + s ) attains its maximum under the conditions n 1 + n + +n t = n s and 1 n 1 n n t if and only if n 1 = n = =n t 1 = 1 and n t = β s + 1. By Claim 4, we get RDD + (G ) = F 1 (n 1, n,, n t ) + (s 1)F (n 1, n,, n t ) +(s + (n 3) + 1)F 3 (n 1, n,, n t ) + 1 F 4(n 1, n,, n t ), which attains its maximum if and only if n 1 = n = = n t 1 = 1 and n t = n s t + 1 = β s + 1. It follows that G = K s + (K β s+1 K n+s β 1 ). Simple calculations shows that RDD + (K β s+1, K β s+1 ) = ( β s+1) (β s), RDD + (K n+s β 1, K n+s β 1 ) = ( n+s β 1) s, RDD + (K β s+1, K n+s β 1 ) = (β s + 1)(n + s β 1)β, RDD + (K s, K s ) = ( s ) (n 1). Taking into account the contributions to the RDD + -value above, it follows that RDD + (G ) = RDD + (K β s+1, K β s+1 ) + RDD + (K n+s β 1, K n+s β 1 ) + RDD + (K β s+1, K n+s β 1 ) + RDD + (K s, K s ) = 7 4n + 4β 1 s3 + s + n 5n 4β 8nβ + 4 s +4β 3 + nβ + (n 1)β. 13

7 J Comb Optim (015) 30: Analyzing the function on s (s) = 7 4n + 4β 1 s3 + s + n 5n 4β 8nβ + 4 s + 4β 3 +nβ + (n 1)β. It follows that s β, since t s = n β t + s β. By taking derivatives, we have (s) = 1 s + (4n + 4β 1)s + n 5n 4β 8nβ + 4 and (s) = 1s + 4n + 4β 1 = 1(β s) + 4n + 3β 1 4n + 3β 1 > 0. This implies that (s) is a strictly convex function for s β, and the maximum value of (s) is attained when s = 1ors = β. (1) = 4β 3 + (n 1)β (3n 11)β + n n 4 (β) = 1 β3 1 β + n 3n + β. After subtraction, we obtain (1) (β) = 7 β3 + 4n 3 Now, let us consider the function on β It follows that (β) = 7 β3 + 4n 3 β n + 3n 0 β n + 3n 0 and β + n n 4. β + n n 4. (β) = 1 β + (4n 3)β n + 3n 0. The quadratic equation (β) = 0 has two distinct roots, since 37n 11n+109 > 0. Let σ be its positive root, namely σ = 3 4n + 37n 11n If β > σ, and (β) > 0, then (β) is an increasing function, and (β) > (1) = 0 follows. This implies that (1) > (β).ifβ<σ,then (1) < (β). This completes the proof of Theorem.3. 13

8 44 J Comb Optim (015) 30: K-decomposition for RDD + Let k be an integer not less than, a k -decomposition D k = (G 1, G,, G k ) of a graph G is a partition of its edge set to form k spanning subgraphs G 1, G,, G k, each of the G i is said to be a cell of G. We encourage the interested reader to refer (Aouchiche and Hansen 013; Nordhaus and Gaddum 1956) for some more information. Li and Zhao (004) introduced the concept of the generalized first Zagreb index for a graph, which was defined as: M 1,ɛ (G) = v V [deg G (v)]ɛ. In particular, M 1, (G) = M 1 (G) is the first Zagreb index. We encourage the interested reader to consult (Ashrafi et al. 011; Došlić and Bonchev 008; Klein et al. 007) for details on these indices. Lemma 3.1 (Su et al. 01b) Let D k be a k-decomposition of the complete graph K n and t be an integer. Then (a) n(n 1) ɛ k 1 ɛ k M 1,ɛ (G i ) n(n 1) ɛ, if ɛ>1; (b) n(n 1) ɛ k M 1,ɛ (G i ) n(n 1) ɛ k 1 ɛ, if 0 <ɛ<1; (c) n(n 1) ɛ k 1 ɛ k M 1,ɛ (G i ) n [ t + t(n ) ɛ], if ɛ<0 and k = t; (d) n(n 1) ɛ k 1 ɛ k M 1,ɛ (G i ) n [ t+(t+1)(n ) ɛ], if ɛ<0 and k = t+1. The following results will be used in our proofs. Lemma 3. (Hua and Zhang 01) Let G be a connected graph of order n and m 1. Then (n 1)m diam(g) + (diam(g) 1)M 1(G) diam(g) RDD + (G) (n 1)m + M 1(G), with either equality if and only if diam(g). Hua and Zhang characterized connected graphs with the maximum and minimum RDD + -value, respectively. More precisely: Lemma 3.3 (Hua and Zhang 01) Among all nontrivial connected graphs of order n, the graphs with the maximum and minimum RDD + -value are K n and P n (the path of order n), respectively. This bring us to the main result in this section. Theorem 3.4 Let D k be a k-decomposition of the complete graph K n. Then krdd + (P n ) k RDD + (G i ) n(n 1), with left equality if and only if each cell G i = Pn, and with right equality if and only if the diameter of G i is at most two. 13

9 J Comb Optim (015) 30: Proof Let m i denotes the size of the i-th cell G i of D k. By Lemmas 3.1 and 3., we get k RDD + (G i ) k = (n 1) [ (n 1)m i + 1 ] M 1(G i ) k m i + 1 n(n 1). k M 1 (G i ) An et al. (011) proved that for any sufficiently large n with respect to k, there is a k-decomposition D k of K n such that diam(g i ) = for each i = 1,,, k. Hence the right equality holds if and only if the diameter of G i is at most two. The lower bound can be verified directly by Lemma 3.3 k RDD + (G i ) krdd + (P n ). This completes the proof of Theorem RDD + -value for operation graphs In this section we present formulas for computing RDD + of join and the Cartesian product of graphs. 4.1 Join graphs The join G 1 + G of graphs G 1 and G with disjoint vertex sets V 1 and V and edge sets E 1 and E is the graph union G 1 G together with all edges joining V 1 and V. The following lemma is crucial in what follows later. For convenience, we use V i and E i to denote the order and the size of graph G i for i = 1,, respectively. Lemma 4.1 Let G 1 and G be two connected graphs. Then we have (a) V (G 1 + G ) = V 1 + V and E(G 1 + G ) = E 1 + E + V 1 V. (b) The join of graphs is associative and commutative. (c) deg G1 +G (u) = deg G1 (u)+ V for u V 1 and deg G1 +G (u) = deg G (v)+ V 1 for v V. (d) 0, if u 1 = u, dist G1 +G (u 1, u )= 1, if u 1 u E 1 or u 1 u E or (u 1 V 1 and u V ),, otherwise. 13

10 444 J Comb Optim (015) 30: Proof The claims (a) (d) are derived from the definitions and some well known results of the book of Imrich and Klavžar (000). Let RDD + + (G i, G i ) (resp., RDD + (G i, G i )) denote the contribution to the RDD + -value by adjacent (resp., non-adjacent) vertices in G i for i = 1,, respectively. Theorem 4. Let G = G 1 + G be the join of two connected graphs G 1 and G. Then RDD + (G 1 +G ) = M 1 (G 1 )+M 1 (G )+ 1 M 1(G 1 )+ 1 M 1(G )+4 V E 1 +4 V 1 E + V 1 V + V 1 V + V E 1 + V 1 E, where M 1 (G) is the first Zagreb co-index of graph G. Proof Simple calculations shows that RDD ++ (G 1, G 1 ) = uv E 1 [ degg1 (u) + deg G1 (v) + V ] = M 1 (G 1 ) + V E 1, RDD ++ (G, G ) = uv E [ degg (u) + deg G (v) + V 1 ] = M 1 (G 1 ) + V 1 E, RDD + (G 1, G 1 ) = uv E 1 [ degg1 (u) + deg G1 (v) + V ] 1 = 1 M 1(G 1 ) + V E 1, RDD + (G, G ) = [ uv E degg (u) + deg G (v) + V 1 ] 1 = 1 M 1(G ) + V 1 E, RDD + (G 1, G ) = [ u V 1 v V degg1 (u) + deg G (v) + V 1 + V ] = V E 1 + V 1 E + V 1 V + V 1 V. Hence, the reciprocal sum-degree distance of G 1 + G can be written as the sum RDD + (G) = RDD + + (G 1, G 1 ) + RDD + + (G, G ) + RDD + (G 1, G 1 ) + RDD + (G, G ) + RDD + (G 1, G ) = M 1 (G 1 ) + M 1 (G ) + 1 M 1(G 1 ) + 1 M 1(G ) + 4 V E 1 +4 V 1 E + V 1 V + V 1 V + V E 1 + V 1 E. This completes the proof of Theorem 4.. Let K s,t be the bipartite graph with two partitions having s and t vertices. Note that K s,t = K s + K t, we have: Corollary 4.3 RDD + (K s,t ) = RDD + (K s + K t ) = st + s t + s ( t ( ) + t s ). 4. Cartesian product graphs The Cartesian product G 1 G of graphs G 1 and G is a graph with vertex set V 1 V, and two vertices (u 1, u ) and (v 1,v ) adjoint by an edge: u 1 = v 1 and u v in G, (u 1, u ) (v 1,v ) or u = v and u 1 v 1 in G 1. 13

11 J Comb Optim (015) 30: Lemma 4.4 Let G 1 and G be two connected graphs. Then we have (a) V (G 1 G ) = V 1 V and E(G 1 G ) = V 1 E + V E 1. (b) deg G1 G (u,v)= deg G1 (u) + deg G (v); (c) dist G1 G ((u, u ), (v,v )) = dist G1 (u,v ) + dist G (u,v ). Proof The claims (a) (c) are derived from the definitions and some well known results of the book of Imrich and Klavžar Imrich and Klavžar (000). Theorem 4.5 Let G = G 1 G be the Cartesian product of graphs G 1 and G. Then RDD + (G 1 G ) V RDD + (G 1 )+ V 1 RDD + (G )+4 E 1 H(G )+4 E H(G 1 ). Proof Let u = (u, u ) and v = (v,v ) be two vertices in G = G 1 G.Bythe definition of RDD + and Lemma 4.4, wehave RDD + (G 1 G ) = {u,v} V (G) deg G1 G (u) + deg G1 G (v) dist G1 G (u,v) deg G1 (u ) + deg G (u ) + deg G1 (v ) + deg G (v ) = dist G1 (u,v ) + dist G (u,v ) {u,v} V (G) [ degg1 (u ) + deg G1 (v ) dist a V 1 {u,v G (u,v + deg G (u ) + deg G (v ] ) ) dist G (u,v ) } V + [ degg1 (u ) + deg G1 (v ) dist b V G1 (u,v + deg G (u ) + deg G (v ] ) ) dist G1 (u,v ) {u,v } V 1 = V RDD + (G 1 ) + V 1 RDD + (G ) + 4 E 1 H(G ) + 4 E H(G 1 ). This completes the proof of Theorem 4.5. Acknowledgments The authors are very grateful to the anonymous referees for their valuable suggestions, corrections and comments that helped to improve the original manuscript. This work has been supported by the National Natural Science Foundation of China (No , ). Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited. References Aouchiche M, Hansen P (013) A survey of Nordhaus Gaddum type relations. Discrete Appl Math 161: Ashrafi AS, Došlić T, Hamzeh A (011) Extremal graphs with respect to the Zagreb co-indices. MATCH Commun Math Comput Chem 65:85 9 An Z, Wu B, Li D, Wang Y, Su G (011) Nordhaus-Gaddum-type theorem for diameter of graphs when decomposing into many parts. Discrete Math Algorithms Appl 3: Bondy JA, Murty USR (1976) Graph theory with applications. Elsevier, New York Caporossi G, Paiva M, Vuki cević D, Segatto M (01) Centrality and betweenness: vertex and edge decomposition of the Wiener index. MATCH Commun Math Comput Chem 68:93 30 Cui Z, Liu B (01) On Harary matrix, Harary index and Harary energy. MATCH Commun Math Comput Chem 68:

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