Bounds for the Kirchhoff Index of Bipartite Graphs

Abstract

A (𝑚,𝑛)-bipartite graph is a bipartite graph such that one bipartition has m vertices and the other bipartition has n vertices. The tree dumbbell 𝐷(𝑛,𝑎,𝑏) consists of the path 𝑃_{𝑛−𝑎−𝑏} together with a independent vertices adjacent to one pendent vertex of 𝑃_{𝑛−𝑎−𝑏} and b independent vertices adjacent to the other pendent vertex of 𝑃_{𝑛−𝑎−𝑏}. In this paper, firstly, we show that, among (𝑚,𝑛)-bipartite graphs (𝑚≤𝑛), the complete bipartite graph 𝐾_𝑚,𝑛 has minimal Kirchhoff index and the tree dumbbell 𝐷(𝑚+𝑛,⌊𝑛−(𝑚+1)/2⌋,⌈𝑛−(𝑚+1)/2⌉) has maximal Kirchhoff index. Then, we show that, among all bipartite graphs of order 𝑙, the complete bipartite graph 𝐾_{⌊𝑙/2⌋,𝑙−⌊𝑙/2⌋} has minimal Kirchhoff index and the path 𝑃_𝑙 has maximal Kirchhoff index, respectively. Finally, bonds for the Kirchhoff index of (𝑚,𝑛)-bipartite graphs and bipartite graphs of order 𝑙 are obtained by computing the Kirchhoff index of these extremal graphs.

1. Introduction

Let 𝐺 be a connected graph with vertices labeled as 𝑣₁,𝑣₂,…,𝑣_𝑛. The distance between vertices 𝑣_𝑖 and 𝑣_𝑗, denoted by 𝑑_𝐺(𝑣_𝑖,𝑣_𝑗), is the length of a shortest path between them. The famous Wiener index 𝑊(𝐺) [1] is the sum of distances between all pairs of vertices, that is𝑊(𝐺)=_𝑖<𝑗𝑑_𝐺𝑣_𝑖,𝑣_𝑗. (1.1)

In 1993, Klein and Randić [2] introduced a new distance function named resistance distance on the basis of electrical network theory. They view 𝐺 as an electrical network 𝑁 such that each edge of 𝐺 is assumed to be a unit resistor. Then, the resistance distance between vertices 𝑣_𝑖 and 𝑣_𝑗, denoted by 𝑟_𝐺(𝑣_𝑖,𝑣_𝑗), is defined to be the effective resistance between nodes 𝑣_𝑖 and 𝑣_𝑗 in 𝑁. Analogous to the Wiener index, the Kirchhoff index 𝐾𝑓(𝐺) [2, 3] is defined as𝐾𝑓(𝐺)=_𝑖<𝑗𝑟_𝐺𝑣_𝑖,𝑣_𝑗. (1.2)

As an analogy to the famous Wiener index, the Kirchhoff index is an important molecular structure descriptor [4], and thus it is well studied in both mathematical and chemical literatures. For more information on the Kirchhoff index, the readers are referred to recent papers [5–16] and references therein.

It is of interest to determine bounds for the Kirchhoff index of some classes of graphs and characterize extremal graphs as well. Along this line, much research work has been done. For a general graph 𝐺, Lukovits et al. [17] proved that 𝐾𝑓(𝐺)≥𝑛−1 with equality if and only if 𝐺 is a complete graph, and they also indicated that the maximal Kirchhoff index graph is the path 𝑃_𝑛. Palacios [18] proved that 𝐾𝑓(𝐺)≤1/6(𝑛³−𝑛) with equality if and only if 𝐺 is a path. For a circulant graph, Zhang and Yang [19] showed that(𝐺)≤𝑛𝑛−1≤𝐾𝑓³−𝑛12, (1.3)

where the first equality holds if and only if 𝐺 is a complete graph and the second does if and only if 𝐺 is a cycle. Furthermore, tight bounds for the Kirchhoff index are also obtained for a special class of unicyclic graphs [20], bicyclic graphs [21, 22], and Cacti [23].

Bipartite graphs are perhaps the most basic of objects in graph theory, both from a theoretical and practical point of view. Let 𝐺 be a bipartite graph with bipartition 𝑋 and 𝑌 such that 𝑋 is the set of white vertices and 𝑌 is the set of black vertices. Suppose that |𝑋|=𝑚 and |𝑌|=𝑛. Such graph is also known as (𝑚,𝑛)-bipartite graph. Without loss of generality, we supposed that 𝑚≤𝑛. The tree dumbbell 𝐷(𝑛,𝑎,𝑏) consists of the path 𝑃_{𝑛−𝑎−𝑏} together with 𝑎 independent vertices adjacent to one pendent vertex of 𝑃_{𝑛−𝑎−𝑏} and 𝑏 independent vertices adjacent to the other pendent vertex of 𝑃_{𝑛−𝑎−𝑏}. For instance, 𝐷(13,3,3) is referred to Figure 1.

Figure 1 

𝐷(13,3,3).

In the next section, we first obtain that 𝐾_𝑚,𝑛 has the minimal Kirchhoff index among all (𝑚,𝑛)-bipartite graphs according to strictly increasing property of the Kirchhoff index. Then we prove that tree dumbbell 𝐷(𝑚+𝑛,⌊(𝑛−𝑚+1)/2⌋,⌈(𝑛−𝑚+1)/2⌉) has maximal Kirchhoff index among all (𝑚,𝑛)-bipartite graphs. Therefore, tight bounds for the Kirchhoff index of (𝑚,𝑛)-bipartite graphs are determined. In the last section, we discuss general bipartite graphs of order 𝑙. We obtain that, among all bipartite graphs of order 𝑙, complete bipartite graph 𝐾_{⌊𝑙/2⌋,𝑙−⌊𝑙/2⌋} and path 𝑃_𝑙 have minimal and maximal Kirchhoff index, respectively. Thus bounds for the Kirchhoff index of bipartite graphs of order 𝑙 are also obtained.

2. (𝑚,𝑛)-Bipartite Graphs with Extremal Kirchhoff Index

Lemma 2.1 (see [19]). Let 𝐺 be a connected graph with 𝑛 vertices and 𝐻 a connected spanning subgraph of 𝐺. Then, 𝐾𝑓(𝐺)≤𝐾𝑓(𝐻) with equality if and only if 𝐺=𝐻.

By Lemma 2.1, the complete bipartite graph 𝐾_𝑚,𝑛 has minimal Kirchhoff index among all (𝑚,𝑛)-bipartite graphs. Now, we compute the Kirchhoff index of 𝐾_𝑚,𝑛.

Lemma 2.2. 𝐾𝑓𝐾_𝑚,𝑛=(𝑚+𝑛−1)𝑚²+𝑛²−𝑚𝑛𝑚𝑛. (2.1)

Proof. For 𝐾_𝑚,𝑛, Klein [24] obtained that the resistance distance between two vertices of different parts is (𝑚+𝑛−1)/𝑚𝑛, the resistance distance between two vertices of 𝑚-vertex part and 𝑛-vertex part is 2/𝑛 and 2/𝑚, respectively. Hence, 𝐾𝑓𝐾_𝑚,𝑛=𝑚𝑛𝑚+𝑛−1+𝑚22𝑚𝑛𝑛+𝑛22𝑚=(𝑚+𝑛−1)𝑚²+𝑛²−𝑚𝑛𝑚𝑛. (2.2)

In the following, we search for (𝑚,𝑛)-bipartite graph with maximal Kirchhoff index. By Lemma 2.1, the graph possesses maximal Kirchhoff index must be a tree since otherwise any of its spanning tree has lager Kirchhoff index than it. It is well known that the Kirchhoff index and the Wiener index concise for trees. Hence, we only need to consider the Wiener index which has been extensively studied. Now, we introduce some well-known results on the Wiener index of trees.

Let 𝑃_𝑛 and 𝑆_𝑛 denote 𝑛-vertex path and 𝑛-vertex star, respectively. Then we have the following.

Lemma 2.3 (see [25]). Let 𝑇 be any 𝑛-vertex tree different from 𝑃_𝑛 and 𝑆_𝑛. Then, 𝑊𝑆_𝑛<𝑊(𝑇)<𝑊𝑃_𝑛. (2.3)

It is also obtained in [25] that𝑊𝑆_𝑛=(𝑛−1)², (2.4)𝑊𝑃_𝑛=3=𝑛+1𝑛³−𝑛6. (2.5)

Let 𝑒=(𝑥,𝑦) be an edge of 𝑇. Let 𝑛₁(𝑒) be the number of vertices of 𝑇 lying closer to 𝑥 than to 𝑦, and let 𝑛₂(𝑒) be the number of vertices of 𝑇 lying closer to 𝑦 than to 𝑥. That is,𝑛₁(𝑒)=𝑣∣𝑣∈𝑉(𝑇),𝑑_𝑇(𝑣,𝑥)<𝑑_𝑇,𝑛𝑣,𝑦₂(𝑒)=𝑣∣𝑣∈𝑉(𝑇),𝑑_𝑇𝑣,𝑦<𝑑_𝑇(𝑣,𝑥). (2.6)

Theorem 2.4 (see [1]). Let 𝑇 be a 𝑛-vertex tree. Then, 𝑊(𝑇)=_{(𝑇)𝑒∈𝐸}𝑛₁(𝑒)𝑛₂(𝑒). (2.7)

In the following, we let 𝑊_𝑇(𝑒)=𝑛₁(𝑒)𝑛₂(𝑒).

Theorem 2.5. 𝐷(𝑚+𝑛,⌊(𝑛−𝑚+1)/2⌋,⌈(𝑛−𝑚+1)/2⌉) has maximal Kirchhoff index among all (𝑚,𝑛)-bipartite graphs.

Proof. Suppose that 𝑇 is the tree possessing maximal Wiener (Kirchhoff) index among all (𝑚,𝑛)-bipartite graphs.
Case 1. 𝑛=𝑚 or 𝑛=𝑚+1. In this case, 𝐷(𝑚+𝑛,⌊(𝑛−𝑚+1)/2⌋,⌈(𝑛−𝑚+1)/2⌉) is the path 𝑃_𝑚+𝑛. By Lemma 2.3, the result holds.Case 2. 𝑛>𝑚+1. Let 𝑃 be a longest path in 𝑇 with end vertices 𝑎 and 𝑏. Suppose that 𝑎^′ and 𝑏^′ are neighbors of 𝑎 and 𝑏 in 𝑃, respectively.Claim 1. The inner vertices of 𝑃 all have degree 2 in 𝑇 except for 𝑎^′ and 𝑏^′.
Suppose to the contrary that there exists an inner point 𝑢 of 𝑃 different from 𝑎^′ and 𝑏^′ has degree lager than 2 and 𝑣 is a neighbor of 𝑢 such that 𝑣∈𝑃. Suppose that the size of the component of 𝑇−𝑢𝑣 containing 𝑣 is 𝑘. Suppose that 𝑒₁ and 𝑒₂ are edges in 𝑃 incident to 𝑢. Let 𝐶_𝑎, 𝐶_𝑏, and 𝐶_𝑢 denote the components of 𝑇−𝑒₁−𝑒₂ containing 𝑎, 𝑏, and 𝑢, respectively. We choose from 𝐶_𝑎 and 𝐶_𝑏 the one containing less vertices, say 𝐶_𝑎. 𝑎 and 𝑎^′ must have one that belongs to the part containing 𝑢, say 𝑎. Let 𝑇^′=𝑇−𝑢𝑣+𝑎𝑣 (see Figure 2). Now we show that 𝑊(𝑇)<𝑊(𝑇^′) by considering the contributions of edges. Obviously, 𝑊_𝑇(𝑢𝑣)=𝑊_𝑇′(𝑎𝑣)=𝑘(𝑚+𝑛−𝑘). Let 𝐸 denote the edge set of 𝐸(𝑇)−𝑢𝑣=𝐸(𝑇^′)−𝑎𝑣, and let 𝑃^′ denote the path 𝑎𝑃𝑢. For 𝑒∈𝐸−𝐸(𝑃^′), 𝑊_𝑇(𝑒)=𝑊_𝑇′(𝑒). For 𝑒∈𝐸(𝑃^′), suppose that 𝐶_𝑎(𝑒) and 𝐶_𝑏(𝑒) are components of 𝑇−𝑒 containing 𝑎 and 𝑏, respectively. Then, 𝑊_𝑇(𝑒)=|𝐶_𝑎(𝑒)||𝐶_𝑏(𝑒)| and 𝑊_𝑇′(𝑒)=(|𝐶_𝑎(𝑒)|+𝑘)(|𝐶_𝑏(𝑒)|−𝑘). Then, 𝑊_𝑇′(𝑒)−𝑊_𝑇(𝑒)=𝑘𝐶_𝑏(𝑒)−𝐶_𝑎(𝑒)−𝑘>𝑘𝐶_𝑏+𝑘−𝐶_𝑎(𝑒)−𝑘≥𝑘𝐶_𝑏−𝐶_𝑎≥0. (2.8) Hence, 𝑊𝑇^′=𝑊_𝑇′(𝑎𝑣)+_{𝑃𝑒∈𝐸−𝐸′}𝑊_𝑇′(𝑒)+_{𝑃𝑒∈𝐸′}𝑊_𝑇′(𝑒)>𝑊_𝑇(𝑢𝑣)+_{𝑃𝑒∈𝐸−𝐸′}𝑊_𝑇(𝑒)+_{𝑃𝑒∈𝐸′}𝑊_𝑇(𝑒)=𝑊(𝑇). (2.9) This contradicts the choice of 𝑇.Claim 2. Both 𝑎 and 𝑏 belong to 𝑌.
Suppose not. Then, we can distinguish the following two cases.Subcase 1. Both 𝑎 and 𝑏 belong to 𝑋. By Claim 1, the inner vertices of 𝑃 all have degree 2 in 𝑇; hence, the vertices of 𝑌 all belong to 𝑃, that is, 𝑛<𝑚, a contradiction.Subcase 2. 𝑎 and 𝑏 belong to different parts. Suppose that 𝑎 belongs to 𝑌. By claim 1 and 𝑛>𝑚+1, we have 𝑑_𝑇(𝑏^′)>𝑑_𝑇(𝑎^′)+1. Let 𝑇^′=𝑇−𝑏^′𝑏+𝑎𝑏 (see Figure 3). Now, we show that 𝑊(𝑇^′)<𝑊(𝑇). Obviously, 𝑊_𝑇′(𝑎𝑏)=𝑊_𝑇(𝑏^′𝑏). Let 𝐸₁ denote the edge set of 𝐸(𝑇)−𝑢𝑣=𝐸(𝑇^′)−𝑎𝑣, and let 𝑃₁ denote the path 𝑎𝑃𝑏^′. For 𝑒∈𝐸₁−𝐸(𝑃₁), 𝑊_𝑇(𝑒)=𝑊_𝑇′(𝑒). Suppose that the edges of 𝑃₁ are 𝑎𝑎^′=𝑒₁,𝑒₂,…,𝑒_𝑙 such that 𝑒_𝑖 is adjacent to 𝑒_𝑖+1 for 1≤𝑖<𝑘. It is easy to see that 𝑊_𝑇′(𝑒₁)=2(𝑚+𝑛−2)>𝑊_𝑇(𝑒₁)=𝑚+𝑛−1 and 𝑊_𝑇′(𝑒_𝑖)=𝑊_𝑇(𝑒_𝑖+1) for 2≤𝑖<𝑘. What is left is to compare 𝑊_𝑇′(𝑒_𝑘) with 𝑊_𝑇(𝑒₂). 𝑊_𝑇′(𝑒_𝑘)=(𝑑_𝑇(𝑏^′)−1)(𝑚+𝑛−𝑑_𝑇(𝑏^′)+1) and 𝑊_𝑇(𝑒₂)=𝑑_𝑇(𝑎^′)(𝑚+𝑛−𝑑_𝑇(𝑎^′)). Then, 𝑊_𝑇′𝑒_𝑘−𝑊_𝑇𝑒₂=𝑑_𝑇𝑏^′−1−𝑑_𝑇𝑎^′𝑚+𝑛−𝑑_𝑇𝑎^′−𝑑_𝑇𝑏^′+1≥0 (2.10) since 𝑑_𝑇(𝑏^′)−1−𝑑_𝑇(𝑎^′)>0 and 𝑚+𝑛−𝑑_𝑇(𝑎^′)−𝑑_𝑇(𝑏^′)+1≥0 with equality if and only if 𝑎^′ and 𝑏^′ are adjacent. Hence, ∑_{𝑒∈𝐸(𝑃1)}𝑊_𝑇′(𝑒)>∑_{𝑒∈𝐸(𝑃1)}𝑊_𝑇(𝑒). Thus, 𝑊𝑇^′=𝑊_𝑇′(𝑎𝑏)+_{𝑒∈𝐸1𝑃−𝐸1}𝑊_𝑇′(𝑒)+_{𝑃𝑒∈𝐸1}𝑊_𝑇′(𝑒)>𝑊_𝑇𝑏^′𝑏+_{𝑒∈𝐸1𝑃−𝐸1}𝑊_𝑇(𝑒)+_{𝑃𝑒∈𝐸1}𝑊_𝑇(𝑒)=𝑊(𝑇). (2.11) As before, this contradicts the choice of 𝑇.Claim 3. The length of 𝑃 is 2𝑚.
By Claims 1 and 2, the vertices of 𝑋 are all contained in 𝑃, the end vertices of 𝑃 are both contained in 𝑌. Hence the length of 𝑃 is 2𝑚 as claimed.Claim 4. |𝑑_𝑇(𝑎^′)−𝑑_𝑇(𝑏^′)|≤1. Suppose to the contrary that |𝑑_𝑇(𝑎^′)−𝑑_𝑇(𝑏^′)|≥2. Without less of generality, suppose that 𝑑_𝑇(𝑏^′)−𝑑_𝑇(𝑎^′)≥2. Let 𝑇^′=𝑇−𝑏^′𝑏+𝑎^′𝑏. We can prove that 𝑊(𝑇^′)>𝑊(𝑇) by methods similar to the proof of Claim 2.
By Claims 1, 2, 3, and 4, we may conclude that 𝑇=𝐷(𝑚+𝑛,⌊(𝑛−𝑚+1)/2⌋,⌈(𝑛−𝑚+1)/2⌉), which implies Theorem 2.5.

(a)

(b)

Figure 2 

𝑇 (a) and 𝑇′ (b) in the proof of Claim 1.

(a)

(b)

Figure 3 

𝑇 (a) and 𝑇′ (b) in the proof of Subcase 2 of Claim 2.

Now, we compute the Kirchhoff (Wiener) index of 𝐷(𝑚+𝑛,⌊(𝑛−𝑚+1)/2⌋,⌈(𝑛−𝑚+1)/2⌉). For convenience, in what follows, we denote 𝐷(𝑚+𝑛,⌊(𝑛−𝑚+1)/2⌋,⌈(𝑛−𝑚+1)/2⌉) by 𝐷^∗.

If 𝑚=𝑛, 𝐷^∗ is the path 𝑃_𝑚+𝑛. Hence, by (2.5), 𝐾𝑓𝐷^∗=𝐾𝑓𝑃_𝑚+𝑛=3𝑚+𝑛+1. (2.12) Otherwise, let 𝐸₁=𝑒∈𝐷^∗∣𝑒isincidenttoaleafof𝐷^∗,𝐸₂=𝐸𝐷^∗−𝐸₁. (2.13) For 𝑒∈𝐸₁, obviously 𝑊_𝐷∗(𝑒)=𝑚+𝑛−1. Noticing that 𝐷^∗ has 𝑛−𝑚+1 leaves, _𝑒∈𝐸1𝑊_𝐷∗(𝑒)=(𝑛−𝑚+1)(𝑚+𝑛−1)=𝑛²−(𝑚−1)². (2.14) We can see that the induced subgraph of 𝐸₂ is the path 𝑃_2𝑚−1, from which 𝐷^∗ can be obtained by adding ⌊(𝑛−𝑚+1)/2⌋ pendant edges to one of its endpoint and ⌈(𝑛−𝑚+1)/2⌉ pendant edges to the other endpoint. Hence, the degrees of endpoints of the path 𝑃_2𝑚−1 in 𝐷^∗ are ⌊(𝑛−𝑚+1)/2⌋+1 and ⌈(𝑛−𝑚+1)/2⌉+1, respectively. Therefore, _𝑒∈𝐸2𝑊_𝐷∗(𝑒)=^{𝑚+𝑛−(⌈(𝑛−𝑚+1)/2⌉+1)}_{⌊⌋𝑖=(𝑛−𝑚+1)/2+1}𝑖(𝑚+𝑛−𝑖). (2.15) Hence, the Kirchhoff index of 𝐷^∗ is 𝐾𝑓𝐷^∗=𝑊𝐷^∗=_𝑒∈𝐸1𝑊_𝐷∗(𝑒)+_𝑒∈𝐸2𝑊_𝐷∗(𝑒)=𝑛²−(𝑚−1)²+^{𝑚+𝑛−(⌈𝑛−𝑚+1/2⌉+1)}_{⌊⌋𝑖=𝑛−𝑚+1/2+1}𝑖()=1𝑚+𝑛−𝑖6−2𝑚+3𝑚²−𝑚³−6𝑚𝑛+6𝑚²𝑛+3𝑛²+3𝑚𝑛²()(),1𝑛−𝑚≡0mod26−3+𝑚+3𝑚²−𝑚³−6𝑚𝑛+6𝑚²𝑛+3𝑛²+3𝑚𝑛²(𝑛−𝑚)≡1(mod2). (2.16)

In sum, we have our main result.

Theorem 2.6. For (𝑚,𝑛)-bipartite graph 𝐺(𝑚≤𝑛), we have (𝑚+𝑛−1)𝑚²+𝑛²−𝑚𝑛(𝐺)≤1𝑚𝑛≤𝐾𝑓6−2𝑚+3𝑚²−𝑚³−6𝑚𝑛+6𝑚²𝑛+3𝑛²+3𝑚𝑛²()(),1𝑛−𝑚≡0mod26−3+𝑚+3𝑚²−𝑚³−6𝑚𝑛+6𝑚²𝑛+3𝑛²+3𝑚𝑛²(𝑛−𝑚)≡1(mod2). (2.17) The first equality holds if and only if 𝐺=𝐾_𝑚,𝑛, and the second does if and only if 𝐺=𝐷(𝑚+𝑛,⌊(𝑛−𝑚+1)/2⌋,⌈(𝑛−𝑚+1)/2⌉).

3. Bipartite Graphs with Extremal Kirchhoff Index

In this section, we consider general bipartite graphs of order 𝑙. By Lemmas 2.1 and 2.3, one can see that the path 𝑃_𝑙 has maximal Kirchhoff index among all bipartite graphs of order 𝑙. The minimal bipartite graph of Kirchhoff index must be min_{1≤𝑚≤⌊𝑙/2⌋}{𝐾_{𝑚,𝑙−𝑚}}. By Lemma 2.2,𝐾𝑓𝐾_{𝑚,𝑙−𝑚}=(𝑙−1)𝑚²+(𝑙−𝑚)²−𝑚(𝑙−𝑚)𝑚()=()𝑙−𝑚𝑙−12𝑚²−2𝑚𝑙+𝑙²𝑚()()()𝑙−𝑚−1=𝑙−12𝑚𝑚−𝑙+𝑙²𝑚()𝑙𝑙−𝑚−1=−2𝑙+1+²𝑚().𝑙−𝑚 (3.1) Hence,min_{⌊⌋1≤𝑚≤𝑙/2}𝐾_{𝑚,𝑙−𝑚}=𝐾_{⌊𝑙/2⌋,𝑙−⌊𝑙/2⌋}. (3.2)

It is easy to compute that𝐾𝑓𝐾_{⌊𝑙/2⌋,𝑙−⌊𝑙/2⌋}=(𝑙−1)𝑙²−2𝑙⌊𝑙/2⌋+2⌊𝑙/2⌋²⌊⌊.𝑙/2⌋(𝑙−𝑙/2⌋) (3.3)

Hence, we have the following result.

Theorem 3.1. For bipartite graph 𝐺 of order 𝑙, we have (𝑙−1)𝑙²−2𝑙⌊𝑙/2⌋+2⌊𝑙/2⌋²⌊⌊(𝐺)≤𝑙𝑙/2⌋(𝑙−𝑙/2⌋)≤𝐾𝑓³−𝑙6. (3.4) The first equality holds if and only if 𝐺=𝐾_{⌊𝑙/2⌋,𝑙−⌊𝑙/2⌋}, and the second does if and only if 𝐺=𝑃_𝑙.

Acknowledgment

This work is supported by NSFC (Grant no. 11126255).