The unicyclic graphs with maximum degree resistance distance

Applied Mathematics and Computation 268 (2015) 859–864

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

The unicyclic graphs with maximum degree resistance distance Jianhua Tu, Junfeng Du, Guifu Su∗ School of Science, Beijing University of Chemical Technology, Beijing 100029, China

a r t i c l e

i n f o

Keywords: Mathematical chemistry Resistance distance Degree resistance distance Unicyclic graphs

a b s t r a c t For a connected graph with order n and size m, the cyclomatic number (=number of independent cycles) is equal to γ = m − n + 1. The graphs with γ = 1 are referred to the unicyclic graphs. In this paper, we characterized completely the unicyclic graphs with n vertices having maximum degree resistance distance. © 2015 Elsevier Inc. All rights reserved.

1. Introduction In this paper we consider undirected, finite and simple graphs only, and use standard notations in graph theory (see [5]). Let G = (V, E ) be a graph, we denote simply the order and size with |V | = n and |E | = m if no ambiguity can arise. The distance (we call it ordinary distance here) between two vertices u and v in G is the length of the shortest path connecting them in G, denoted by dG (u, v). The degree of a vertex u ∈ V is the number of edges incident to u, denoted by dG (u). For a connected graph, the cyclomatic number (=number of independent cycles) is equal to γ = m − n + 1. Recall that graphs with γ = 0, 1, 2, 3 are referred to as trees, unicyclic graphs, bicyclic graphs and tricyclic graphs, respectively. Amount of contributions on bicyclic graphs and tricyclic have been obtained, for example in [27,28]. Topological indices are real numbers associated with chemical structures derived from their hydrogen-depleted graphs as a tool for compact and effective description of structural formulas which are used to study and predict the structure–property correlations of organic compounds. There are many publications on topological indices, see [4,10,14,23,29–31]. The sum of the ordinary distances between all pairs of vertices in a graph G is the Wiener index, namely

W (G) =



dG (u, v),

{u,v}⊆V (G)

which was first time introduced by Wiener more than 60 years ago [35]. Initially, the Wiener index was considered as a molecularstructure descriptor used in chemical applications, but soon it attracted the interest of pure mathematicians; for details and additional references see the review [11] and recent papers [8,12,13,16,21,26,32,37]. Eventually, a number of modifications of the Wiener index were proposed, the degree distance is such a graph invariant:

D(G) =



(dG (u) + dG (v))dG (u, v).

{u,v}⊆V (G)

In 2006, Yuan and An [40] presented the maximum value of degree distance for unicyclic graphs. Two years later, Tomescu determined the unicyclic and bicyclic graphs with minimum degree distance in referee [34]. We encourage the readers who are interested in the invariant to consult [6,9,22,34,40] for more information and details. ∗

Corresponding author. E-mail addresses: [email protected] (J. Tu), [email protected] (J. Du), [email protected], [email protected] (G. Su).

http://dx.doi.org/10.1016/j.amc.2015.06.063 0096-3003/© 2015 Elsevier Inc. All rights reserved.

860

J. Tu et al. / Applied Mathematics and Computation 268 (2015) 859–864 Table 1 A family of distance- and degree-based graph invariants. dG (u, v) RG (u, v)

W(G) Kf(G)

D(G) ?

2

Pn−2 1

K3

2

3

n−1

n−2

3 Fig. 1. The funnel Fn with n vertices.

Klein and Randic´ [24] stated that: if fixed resistors in the electrical networks are assigned to edges of a connected graph, then the effective resistance between pairs of vertices is the graphical distance. The resistance distance between two vertices u and v of graph G, denoted by RG (u, v), is originally defined to be the effective resistance between the corresponding two nodes u and v in the electrical network. More results and devolvement can be found in [1–3,25,36,42]. If the ordinary distance is replaced by resistance distance in the expression for the Wiener index, one arrives at the Kirchhoff index:



K f (G) =

RG (u, v).

{u,v}⊆V (G)

In recent decades and especially in recent years, this invariant has been extensively studied in mathematical, physical and chemical aspects (see details from [15,19,38,39,41–43]). Form Table 1, it is immediately seen that one such graph invariant is missing. In 2012, Gutman et al. proposed the concept of the degree resistance distance [20]:



DR (G) =

(dG (u) + dG (v))RG (u, v).

{u,v}⊆V (G)

This quantity is sometimes referred to as the “degree Kirchhoff index” (see [17,18]) and the “additive degree-Kirchhoff index” (see [7,33]). In [20] some properties of DR -index were presented and the unicyclic graphs with the minimum and the second minimum DR -value for unicyclic graphs were also completely characterized. To the best of our knowledge, the maximum value of degree resistance distance for unicyclic graphs has not been considered so far. In this paper, we proved the following: Theorem 1.1. Let G be a unicyclic graph of order n. Then DR (G)  13 (2n3 − 28n + 54). The equality holds if and only if G ∼ = Fn , where Fn is the funnel obtained from K3 and the path Pn−2 by identifying a vertex of K3 with one pendent vertex of Pn−2 , depicted in Fig. 1. The proof of Theorem 1.1 will be given in Section 3. 2. Two transformations We begin with some additional notations. In what follows, for the sake of brevity, we sometimes use u ∈ G to denote u ∈ V(G). For any vertex v in G, we define the following two functions:

K fv (G) =



RG (u, v)

and Dv (G) =

u∈G



dG (u)RG (u, v).

u∈G

The degree resistance distance DR (G) can be represented as

DR (G) =

 v∈G

dG (v)



RG (u, v).

u∈G

Gutman et al. presented the following exact formulas for cycles. Lemma 2.1 ([20]). Let Ck be the cycle with order k and v ∈ V(Ck ). Then k3 −k k3 −k 12 and DR (Ck ) = 3 ; 2 k2 −1 = 6 and Dv (Ck ) = k 3−1 .

(1) K f (Ck ) = (2) K fv (Ck )

(1)

J. Tu et al. / Applied Mathematics and Computation 268 (2015) 859–864

uk Q k

C u1

v

C

Qi

ui

u1

v

u2 Q 2

uk

Qk

ui

Qi

u2

Q2

u

T

861

Fig. 2. The transformation G → Gi .

w2

C

v

w2 u

T

C

w1

v w1

Fig. 3. The transformation G → G .

Let G − u denote the graph obtained by removing u and all edges incident to u from G. A cut vertex of a graph G is a vertex v such that c(G − v) > c(G), where c(G) denotes the number of components of G. Lemma 2.2 ([24]). Let G be a graph, x be a cut vertex of G and let u, v be vertices belonging to different components which arise upon deletion of x. Then RG (u, v) = RG (u, x) + RG (x, v). The following are two auxiliary transformations, which (strictly) increase the Dv -value. Transformation I. Let v be a cut vertex with dG (v) > 1 not belonging to the unique cycle C in the unicyclic graph G (depicted in Fig. 2). Let ui , i ≥ 2, be the unique vertex which is adjacent to v in each component Qi of G − v. Denote by Gi = G − vu1 + u1 ui the new unicyclic graph, then Dv (G) < Dv (Gi ).   By the definition, Dv (G) = w∈G dG (w)RG (w, v) and Dv (Gi ) = w∈G dGi (w)RGi (w, v). Now, we investigate every vertex w i  other than v. When w ∈ kl=2 V (Ql ) \ {ui }, dGi (w) = dG (w) and RGi (w, v) = RG (w, v); when w = ui , dGi (w) = dG (w) + 1 and RGi (w, v) = RG (w, v) = 1; when w ∈ V(C), dGi (w) = dG (w) and RGi (w, v) = RG (w, v) + 1. Thus Dv (G) < Dv (Gi ). k By using analogous approach, we can construct a unicyclic graph sequences {Gt }t=2 such that Gt = Gt−1 − vut + ut ut+1 . Easily to verify that the Dv -value of Gt is a strictly increasing function on t ∈ [2, k − 1].

Transformation II. Let w1 , v, w2 be three successive vertices lying in the unique cycle C (|C | = l) of the unicyclic graph G. Let G = G − vw1 + w1 w2 , then Dv (G) < Dv (G ) (depicted in Fig. 3). In fact, the resulting graph G is also a unicycle graph. Let w be any vertex lying in C such that dG (w, v) = d = 0, ) 1 −1 ) = d(l−d and RG (w, v) = RG (w, w2 ) + RG (w2 , v) = RG (w, w2 ) + 1. Notice that RG (w, w2 ) = ( 1d + then RG (w, v) = ( 1d + l−d l

) l−d) l−d) d d = d(l−d−1 = d(l−1 − l−1 , we have RG (w, v) = RG (w, w2 ) + 1 = d(l−1 − l−1 + 1 > RG (w, v). The inequality holds since l−1 l − 1 − d > 0. By using analogous approach and Lemma 2.2, we get that RG (u, v) ≥ RG (u, v) and dG (u) = dG (u) for any vertex u = w2 in G. This means that the transformation G → G increases the Dv -value. 1 )−1 l−d−1

3. Proof of Theorem 1.1 We first list or prove several lemmas which will be used in later proof. Lemma 3.1 ([15]). Let G be a connected graph of order n, v be a pendant vertex of G and w be its neighbor. Then K fv (G) = K fw

(G − v) + n − 1.

Recall that RG (x, y) = RG (y, x) and RG (x, y) ≥ 0 with equality if and only if x = y [24]. Lemma 3.2. Let G be a unicyclic graph of order n, v be a pendant vertex of G and w be its neighbor. Then Dv (G) = Dw (G − v) + 2n − 1. Proof. From the definition, we have

Dv (G) =



dG (u)RG (u, v)

u∈G

=



dG (u)RG (u, v) + dG (v)RG (v, v)

u∈G−v

=



u∈G−v

dG (u)[RG (u, w) + RG (w, v)] (by RG (v, v) = 0 and Lemma 2.2)

862

J. Tu et al. / Applied Mathematics and Computation 268 (2015) 859–864



=

dG (u)RG (u, w) +

u∈G−v



=



dG (u) (by RG (w, v) = 1)

u∈G−v

dG−v (u)RG−v (u, w) + 2n − 1 (by |E (G)| = n and dG (v) = 1)

u∈G−v

= Dw (G − v) + 2n − 1. This completes the proof.  Lemma 3.3. Let G be a unicyclic graph of order n. Then for any vertex v ∈ V(G), we have K fv (G) ≤

n2 2

−

n 2

− 53 .

Proof. By induction on n. Note that the only unicyclic graph G with three vertices is K3 , then K fv (K3 ) = 4/3 = 9/2 − 3/2 − 5/3 holds for any vertex v ∈ V(K3 ), which can be verified straightforwardly. Assume that the conclusion holds for graphs of order at most n − 1. We distinguish the following two cases. Case 1. The vertex v is a pendant vertex. Let w be the neighbor of v. Clearly G − v satisfies the induction hypothesis, and therefore together with Lemma 3.1 we have

K fv (G) = K fw (G − v) + n − 1

(n − 1)2

≤

2

−

n−1 5 − +n−1 2 3

n2 n 5 − − . 2 2 3

=

Case 2. The vertex v is not a pendant vertex. Then, the degree of v is at least 2. It follows that there are at least 2 vertices with resistance distance 1 from v, and the resistance distance between v and each arbitrary remaining vertex is respectively at most 2, 3, . . . , n − 4, n − 4 + 23 and n − 4 + 23 , with equality if and only if G ∼ = Fn . Note that the Kfv -value will decrease by adding extra edges to the original graph (see details in [15]). Hence



K fv (Fn ) = 1 + 1 + 2 + 3 + · · · + (n − 4) + 2 × n − 4 +

2 3



3n 1 n2 − − 2 2 3 n 5 n2 − − . < 2 2 3

=

This completes the proof.  The following is an analogous result for Dv -function of graphs. Lemma 3.4. Let G be a unicyclic graph of order n. Then for any vertex v ∈ V(G), we have Dv (G) ≤ n2 −

19 3 .

Proof. By induction on n. The only unicyclic graph G with 3 vertices is K3 , and a simple calculation shows that Dv (K3 ) = 8/3 = 9 − 19/3 for any vertex v ∈ V(K3 ). Assume that the conclusion holds for graphs of order at most n − 1. We distinguish the following two cases. If v is a pendant vertex. Let w be the neighbor of v. By using the induction hypothesis and Lemma 3.2, it yields

Dv (G) = Dw (G − v) + 2n − 1 ≤

  19 19 . + 2n − 1 = n2 − (n − 1)2 − 3

3

If v is not a pendant vertex. We need to consider two cases: (1) if v does not belong to the unique cycle C, then v will be a pendent vertex after using Transformation I, which yields that Dv (G) ≤ n2 − 19 3 ; (2) if v lies in the cycle C, by using Transformation II, one can obtain a new unicyclic graph containing a cycle C (|C | < |C|) such that the vertex v is not belonging to the cycle C . Thus the discussion will go back to case (1). Hence, Dv (G) ≤ n2 − 19 3 holds. We have completed the proof.  Lemma 3.5. Let G be a unicyclic graph. If v is a pendant vertex of G and w be the neighbor of v. Then DR (G) = DR (G − v) + Dw

(G − v) + 2K fw (G − v) + 3n − 2.

Proof. The degree resistance distance DR (G) =

DR (G) =



u∈G−v

=

 u∈G−v

dG (u)



RG (u, x) + dG (v)

x∈G

dG (u)



 x∈G−v

u∈G

 x∈G

dG (u)

 x∈G

RG (u, x) can be computed as follows:

RG (v, x)

(2)



RG (u, x) + RG (u, v)

+ K fv (G)

(3)

J. Tu et al. / Applied Mathematics and Computation 268 (2015) 859–864

=





u∈G−v

=





dG (u)

RG−v (u, x) + RG (u, v)

x∈G−v

dG (u)

u∈G−v



863



RG−v (u, x) +

x∈G−v



+ K fv (G)

(4)

dG (u)RG (u, v) + K fv (G)

(5)

u∈G−v

Denote by (u, v) the sum of the first and second terms in Eq. (5), i.e.,

(u, v) =





dG (u)

u∈G−v



RG−v (u, x) +

x∈G−v

dG (u)RG (u, v).

u∈G−v

It is sufficient to compute (u, v). By simple calculations, we get



(u, v) = DR (G − v) + K fw (G − v) +

dG (u)RG (u, v)

u∈G−v



= DR (G − v) + K fw (G − v) +

dG (u)(RG (u, w) + RG (w, v))

u∈G−v



= DR (G − v) + K fw (G − v) +

dG−v (u)RG−v (u, w) +

u∈G−v



dG (u).

u∈G−v

After substituting (u, v) back into Eq. (5), we get

DR (G) = DR (G − v) + K fw (G − v) + Dw (G − v) + 2n − 1 + K fw (G − v) + n − 1 = DR (G − v) + Dw (G − v) + 2K fw (G − v) + 3n − 2. This completes the proof.  In the rest of the present section, we devote ourselves to the proof of our main result. Proof of Theorem 1.1. We shall prove the theorem by induction on n. If n = 3, simple calculation shows that DR (K3 ) = 8 = 1 3 3 (2 × 3 − 28 × 3 + 54). It is clear that K3 is also a funnel F3 , the result holds. In the following we assume that the theorem holds for all n ∈ [3, k − 1]. Let G be a unicyclic graph of order k − 1, then DR (G)  1 3 ∼ 3 (2(k − 1) − 28(k − 1) + 54) and the equality holds if and only if G = Fk−1 . For simplicity, we distinguish the following two cases for n = k. Case 1. There exists a pendant vertex in G. Let v be a pendant vertex in G and let w be its neighbor. By the induction hypothesis, we get

DR (G) = DR (G − v) + Dw (G − v) + 2K fw (G − v) + 3n − 2 (by Lemma 3.5)   1 19 ≤ (2(k − 1)3 − 28(k − 1) + 54) + (k − 1)2 − 3 3



+2 =

(k − 1)2 2

−

(k − 1) 2

5 − 3



+ 3k − 2 (by Lemmas 3.4 and 3.3)

1 (2k3 − 28k + 54). 3

)2 ) In addition, if the equality holds, we must have K fw (G − v) = (k−1 − (k−1 − 53 . Lemma 3.3 yields that w must be the unique 2 2 pendent vertex in Fk−1 . It shows that G ∼ = Fk .

Case 2. There is no pendant vertex in G. We immediately conclude that G = Ck , Lemma 2.1 yields that DR (Ck ) = 28k + 54). We have completed the proof of Theorem 1.1. 

k3 −k 3

<

1 3 3 (2k

−

4. Conclusion In this paper, we completely characterized the unicyclic graphs with maximum degree resistance distance. While our result just determines the maximum value of this topological index for a class of graphs, it suggests further studies on lager class of graphs or general graphs. For example, can one determine the maximum or minimum degree resistance distance for tricyclic graphs or any general graphs? We will investigate these problems as well as others not yet thought of in our further researches. Acknowledgments The authors would like to thank two anonymous referee for a number of helpful and valuable comments and suggestions which led the improvement of this paper. Jianhua Tu was supported by the National Natural Science Foundation of China (no. 11201021) and Beijing Higher Education Young Elite Teacher Project (no. YETP0517).Guifu Suwas supported by China Postdoctoral Science Foundation (no. 2015M570025) and the Fundamental Research Funds for the Central Universities (no. ZY1529).

864

J. Tu et al. / Applied Mathematics and Computation 268 (2015) 859–864

References ´ D.J. Klein, I. Lukovits, S. Nikolic, ´ N. Trinajstic, ´ Resistance-distance matrix: a computational algorithm and its application, Int. J. Quantum Chem. 90 [1] D. Babic, (2002) 166–176. [2] R.B. Bapat, I. Gutman, W.J. Xiao, A simple method for computing resistance distance, Z. Naturforsch. 58a (2003) 494–498. [3] D. Bonchev, A.T. Balaban, X. Liu, D.J. Klein, Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances, Int. J. Quantum Chem. 50 (1994) 1–20. [4] A. Behtoei, M. Jannesari, B. Taeri, Maximum zagreb index, minimum hyper-wiener index and graphs connectivity, Appl. Math. Lett. 22 (2009) 1571–1576. [5] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, Macmillan/Elsevier, London/New York, 1976. [6] O. Bucicovschi, S.M. Cioadˇa, The minimum degree distance of graphs of given order and size, Discr. Appl. Math. 156 (2008) 3518–3521. [7] M. Bianchi, A. Cornaro, J.L. Palacios, A. Torriero, New upper and lower bounds for the additive degree-kirchhoff index, Croat. Chem. Acta 86 (2013) 363–370. [8] Y. Chen, X. Zhang, On wiener and terminal wiener indices of trees, MATCH Commun. Math. Comput. Chem. 70 (2013) 591–602. [9] P. Dankelmannn, I. Gutman, S. Mukwembi, H.C. Swart, On the degree distance of a graph, Discr. Appl. Math. 157 (2009) 2773–2777. [10] K.C. Das, I. Gutman, B. Furtula, Survey on geometric-arithmetic indices of graphs, MATCH Commun. Math. Comput. Chem. 65 (2011) 595–644. [11] A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math. 66 (2001) 211–249. [12] A.A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math. 72 (2002) 247–294. [13] C.M. da Fonseca, M. Ghebleh, A. Kanso, D. Stevanovic, Counterexamples to a conjecture on wiener index of common neighborhood graphs, MATCH Commun. Math. Comput. Chem. 72 (2014) 333–338. [14] M. Dehmer, M. Grabner, The discrimination power of molecular identification numbers revisited, MATCH Commun. Math. Comput. Chem. 69 (2013) 785– 794. [15] L. Feng, G. Yu, K. Xu, Z. Jiang, A note on the kirchhoff index of bicyclic graphs, Ars Comb. 114 (2014) 33–40. [16] B. Furtula, I. Gutman, H. Lin, More trees with all degrees odd having extremal wiener index, MATCH Commun. Math. Comput. Chem. 70 (2013) 293–296. [17] L. Feng, G. Yu, W. Liu, Further results regarding the degree kirchhoff index of graphs, Miskolc Math. Notes 15 (2014) 97–108. [18] L. Feng, I. Gutman, G. Yu, Degree Kirchhoff index of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 69 (2013) 629–648. [19] Q. Guo, H. Deng, D. Chen, The extremal kirchhoff index of a class of unicyclic graphs, MATCH Commun. Math. Comput. Chem. 61 (2009) 713–722. [20] I. Gutman, L. Feng, G. Yu, Degree resistance distance of unicyclic graphs, Trans. Comb. 01 (2012) 27–40. [21] A. Hamzeh, A. Iranmanesh, T. Reti, I. Gutman, Chemical graphs constructed of composite graphs and their q-wiener index, MATCH Commun. Math. Comput. Chem. 72 (2014) 807–823. ´ S. Klavžar, D. Stevanovic, ´ Calculating the degree distance of partial hamming graphs, MATCH Commun. Math. Comput. Chem. 63 (2010) 411–424. [22] A. Ilic, [23] V. Kraus, M. Dehmer, M. Schutte, On sphere-regular graphs and the extremality of information-theoretic network measures, MATCH Commun. Math. Comput. Chem. 70 (2013) 885–900. ´ Resistance distance, J. Math. Chem. 12 (1993) 81–95. [24] D.J. Klein, M. Randic, [25] D.J. Klein, Graph geometry, graph metrics, and winner, MATCH Commun. Math. Comput. Chem. 35 (1997) 7–27. [26] M. Knor, B. Luzar, R. Skrekovski, I. Gutman, On wiener index of common neighborhood graphs, MATCH Commun. Math. Comput. Chem. 72 (2014) 321–332. [27] S. Ji, X. Li, Y. Shi, The extremal matching energy of bicyclic graphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 697–706. [28] X. Li, Y. Shi, M. Wei, J. Li, On a conjecture about tricyclic graphs with maximal energy, MATCH Commun. Math. Comput. Chem. 72 (2014) 183–214. [29] X. Li, Y. Li, Y. Shi, I. Gutman, Note on the homo-lumo index of graphs, MATCH Commun. Math. Comput. Chem. 70 (2013) 85–90. [30] B. Liu, I. Gutman, On general randic indices, MATCH Commun. Math. Comput. Chem. 58 (2007) 147–154. [31] B. Liu, Z. You, A survey on comparing zagreb indices, MATCH Commun. Math. Comput. Chem. 65 (2011) 581–593. [32] H. Lin, Extremal wiener index of trees with given number of vertices of even degree, MATCH Commun. Math. Comput. Chem. 72 (2014) 311–320. [33] J.L. Palacios, Upper and lower bounds for the additive degree-kirchhoff index, MATCH Commun. Math. Comput. Chem. 70 (2013) 651–655. [34] A.I. Tomescu, Unicyclic and bicyclic graphs having minimum degree distance, Discr. Appl. Math. 156 (2008) 125–130. [35] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69 (1947) 17–20. [36] W. Xiao, I. Gutman, On resistance matrices, MATCH Commun. Math. Comput. Chem. 49 (2003) 67–81. [37] K. Xu, M. Liu, K.C. Das, I. Gutman, B. Furtula, A survey on graphs extremal with respect to distance-based topological indices, MATCH Commun. Math. Comput. Chem. 71 (2014) 461–508. [38] Y. Yang, X. Jiang, Unicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 60 (2008) 107–120. [39] Y.J. Yang, H.P. Zhang, Some rules on resistance distance with applications, J. Phys. A: Math. Theor. 41 (2008).445203 (12 pp) [40] H. Yuan, C. An, The unicyclic graphs with maximum degree distance, J. Math. Study 39 (2006) 18–24. [41] H. Zhang, X. Jiang, Y. Yang, Bicyclic graphs with extremal Kirchhoff index, MATCH Commun. Math. Comput. Chem. 61 (2009) 697–712. ´ On resistance-distance and kirchhoff index, J. Math. Chem. 46 (2009) 283–289. [42] B. Zhou, N. Trinajstic, ´ The kirchhoff index and matching number, Int. J. Quantum Chem. 109 (2009) 2978–2981. [43] B. Zhou, N. Trinajstic,