Research Article | Open Access
Some Bounds for the Kirchhoff Index of Graphs
The resistance distance between two vertices of a connected graph is defined as the effective resistance between them in the corresponding electrical network constructed from by replacing each edge of with a unit resistor. The Kirchhoff index of is the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.
Let be a connected graph with vertices labeled as . It is natural to view as an electrical network by imagining each edge of to be a (unit) resistor. In this guise, it is reasonable to consider the effective resistance between any two vertices of , and the novel concept of resistance distance  between any two vertices and of is thus defined as the effective resistance between them. Compared to the (shortest-path) distance between and in , it is well known that with equality if and only if and are connected by unique path .
There are many distance-based molecular structure descriptors, as reviewed in , which have played important roles in QSAR and QSPR. Among these structure descriptors, the most famous one is the Wiener index , which is known as the sum of distances between all pairs of vertices. Analogous to the Wiener index, the Kirchhoff index of [1, 4], denoted by , is defined as the sum of resistance distances between all pairs of vertices; that is,
As summarized in , much work has been done by many researchers to investigate bounds for the Kirchhoff index. There are not only general bounds that are given in terms of various graph structural parameters like the number of vertices, the number of edges, the matching number, the chromatic number, the maximum degree, and the number of spanning trees [5–20], but also bounds for some special interesting classes of graphs, such as circulant graphs, unicyclic graphs, and bicyclic graphs [21–27]. Along this line, we consider the relation between the Kirchhoff index and the independence number as well as the clique number, and bounds are obtained for the Kirchhoff index of graphs via the two graph invariants. In addition, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated. For more information on the Kirchhoff index of graphs, the readers are referred to the most recent papers [28–36] and references therein.
2. General Bounds
2.1. A Lower Bound via the Independence Number
We first introduce some notations. Denote the vertex set and edge set of , respectively, by and . A subset of is called independent if its vertices are mutually nonadjacent. The independence number is the largest cardinality among all independent sets of . The clique number of is the largest set of mutually adjacent vertices in . The degree of vertex , denoted by , is the number of neighbors of . The adjacency matrix of is an matrix with the -entry equal to 1 if vertices and are adjacent and 0 otherwise. Let be the diagonal matrix of vertex degrees. Then the Laplacian matrix of is . Let be the eigenvalues of , called the Laplacian eigenvalues of . Since is connected, and for . The spectrum of , also known as the Laplacian spectrum of , is
The join of and , denoted by , is the graph obtained from by adding all edges between vertices of and that of , where denote the disjoint union. The Laplacian eigenvalues of are characterized in the following result.
Lemma 2 (see ). Let and be graphs such that and . Suppose that and are Laplacian eigenvalues of and . Then the Laplacian eigenvalues of are , , , , , .
The nonincreasing property of the Kirchhoff index, as stated blow, plays an important role in estimating bounds for the Kirchhoff index.
Lemma 3 (see ). Let be a noncomplete graph. If is obtained from by adding an edge, then .
For simplicity, if there is no confusion, we always abbreviate , , , , and to , , , , and , respectively. Throughout the paper, we use , to denote the complete and path graph of order , respectively. We use to denote the complement of . Then the main result of this subsection is given as follows.
Theorem 4. Let be a connected graph with vertices and independence number . Then with equality if and only if .
Proof. Let be a graph having the minimum Kirchhoff index among all connected graphs with vertices and independence number . Then by the nonincreasing property of the Kirchhoff index as given in Lemma 3, it is easily seen that . Now we compute the Kirchhoff index of . Since it is well known that and , by Lemma 2, we get Then according to Theorem 1, it follows that
2.2. An Upper Bound via the Clique Number
Let be the set of graphs with vertices and clique number . Let denote the graph obtained by identifying one end vertex of path with any vertex of . In the following, we have shown that, among all the graphs in , has the maximum Kirchhoff index. To this end, we need the following two lemmas.
Lemma 5 (see ). Let be a graph with a cut edge and and the components of containing and , respectively. Then where denote the sum of resistance distances between and all the other vertices of .
Lemma 6 (see ). Let be an vertex tree different from and . Then
Theorem 7. Let be different from . Then
Proof. If , then is a path. Noticing that the Kirchhoff index is equal to the Wiener index for trees, by Lemmas 3 and 6, the assertion holds. In the following, we assume that .
Let be the graph with the maximum Kirchhoff index and let be the subgraph of such that .
Claim 1 (every component of is connected to by only one edge). Suppose to the contrary that there exists a component of such that it is connected to with edges . Let be the graph obtained from by the deletion of . Then it is easily seen that and is a proper spanning subgraph of . Thus by Lemma 3, we have , contradicting the property that has the maximum Kirchhoff index.
Claim 2 (every component of is a tree). Suppose to the contrary that there exists a component of such that is not a tree. Let be a spanning tree of . Remove from all the edges in but not in to obtain . Then and is a proper subgraph of . Again by Lemma 3, we have , a contradiction.
Claim 3 (every component of is a path, which connects to via an end vertex). Let be any component of . By Claims 1 and 2, we know that is a tree and is connected to by a single edge (). Clearly has two components, one is , and denote the other one by . Then by Lemma 5, Since has the maximum Kirchhoff index and by Lemma 6 and the fact that the Kirchhoff index is equal to the Wiener index for trees, we know that is maximized if and only if is a path and is an end vertex of ; the claim holds.
Claim 4 (every vertex of has degree at most in ). Suppose to the contrary that there exists such that the degree of in is larger than . Then by Claim 1, is connected to at least two components of . Let and be two such components connecting to by edges and , respectively. By Claim 3, both and are paths with and being end vertices of them. Without loss of generality, suppose that the length of is less than or equal to the length of . Let (resp., ) be the end vertex of (resp., ) different from (resp., ), and let be the unique neighbor of . Now construct a new graph from by first deleting the edge and then adding a new edge . Clearly . We show so that Claim 4 is proved by contradiction. It is easily seen that, for any two vertices , distinct from , Thus to show , we need only to show that . On one hand, it is easily verified that On the other hand, for any vertex that is not contained in (), we have Hence it follows that , as required.
Claim 5 ( has only one component). Suppose to the contrary that has at least two components and . Then by Claim 4, and must be connected with via different vertices of , say and . Let and be the two edges connecting with and , respectively. Then both and are paths with and being their end vertices. Suppose that the lengths of and are and , respectively. Without loss of generality, we may assume that . Let (resp., ) be the other end vertex of (resp., ) different from (resp., ), and let be the unique neighbor of . Construct a new graph from by first deleting the edge and then adding a new edge between and . Now we show , which thus gives the desired contradiction. For any two vertices distinct from , it is easily seen that Thus it suffices to show that . On one hand, while which implies that On the other hand, for any vertex that is not contained in , we have Hence it follows that as required.
From Claims 1–5, we deduce that , as desired.
By Lemma 5, simple calculation leads to Consequently, the upper bound on the Kirchhoff index is given in terms of the clique number.
Theorem 8. Let be a connected graph with vertices and clique number . Then with equality holds if and only if .
3. Planar Graphs
A planar graph is a graph which can be drawn in the plane without edges crossing. In this section, we investigate bounds for Kirchhoff index of planar graphs. The following lemma is used.
Lemma 9 (see ). if and only if .
Theorem 10. For an vertex planar graph , The first equality holds if and only if , and the second does if and only if is a path.
Proof. The upper bound is well known and we suffice to show the lower bound. Bearing in mind that, for a planar graph , , together with the fact that , by Cauchy-Schwarz inequality, we have The equality holds if and only if , that is, , which implies by Lemma 9 that is a complete graph. Then the proof is completed by noticing that only and are planar complete graphs for .
Though the lower bound is not sharp for , it can be shown that, up to a scale factor, the bound is asymptotically attainable. One example is the star graph , which has Kirchhoff index . This indicates that the lower bound can be asymptotically attained up to a scale factor of at least . In fact, the scale factor could be improved to at least . For example, consider the planar graph . Since the Laplacian eigenvalues of are  by Lemma 2, we readily obtain that the Laplacian eigenvalues of are Therefore, by Theorem 1, we have Now we consider the asymptotic behavior of as : Hence grows as as , and the scale factor is improved to at least .
4. Fullerene Graphs
A fullerene graph is a cubic 3-connected planar graph with exactly 12 pentagons and other hexagons. Fullerene graphs are well studied in both mathematical and chemical literatures. To give bounds for fullerene graphs, we need some preparations.
The famous Foster first formula, given by Foster , states that where means and are adjacent. Foster’s second formula , also given by Foster, perhaps less well known, states that where is measured across the end vertices of two adjacent edges and and the sum is taken over all adjacent edges. Palacios  extended Foster’s first and second formulae and obtained the so-called Foster third formula, which states that where the sum is taken over all pairs of vertices and such that is a 3-walk. In particular, if is regular, then the above equation can be simply written as where is the number of 3-walks from to .
Now we give lower bounds for resistance distance between any two nonadjacent vertices in in terms of the distance between them.
Lemma 11. Let be a fullerene graph and let . Suppose that . Then
Proof. If , by the inequality , we know that , as desired. Now suppose that . We distinguish the following two cases.
Case 1 ( is even). We partition vertices in into the following parts. For , let and let be the set of the remaining vertices. For , contract into a single vertex . For , contract into a single vertex . Then we obtain a path of length with multiple edges; According to the structure of , it is easily seen that, for , , . For , since every vertex in has at least on neighbor in , the number of edges connecting and is no more than . Similarly, for , the number of edges connecting and is no more than , and the number of edges connecting and is no more than . Hence, according to the series and parallel connection rules of resistors, we know that Then by Rayleigh’s short-cut principle , which states that shorting certain vertices together can only decrease the resistance distances between two given vertices, whereas cutting certain edges can only increase the resistance distance between two given vertices, it follows that
Case 2 ( is odd). First partition vertices in in the following way. For , let for , let and let be the set of remaining vertices. Contract to a single vertex and contract to a single vertex , . Using the same argument as the proof of Case 1, we could obtain that .
The following result is useful.
Lemma 12 (see ). There are exactly pairs of vertices at distance 3 in .
Next we introduce a classical result in graph theory—Menger’s Theorem.
Theorem 13 13 (see , Menger’s Theorem). Let be an undirected graph, and let and be nonadjacent vertices in . Then, the maximum number of pairwise internally disjoint paths in equals the minimum number of vertices from whose deletion separates and .
Now we are ready for our main result.
Theorem 14. For an vertex fullerene graph , one has
Proof. We first prove the lower bound. By Foster’s first formula,
Since is triangle free, it is obvious that the end vertices of any two adjacent edges in are at distance 2. Hence by Foster’s second formula,
Since there are exactly pairs of vertices at distance 3 and by Lemma 11 we know that for , hence
Since for , by Lemma 11, , we have
For the upper bound, we consider any two nonadjacent vertices and . Since is 3-connected, by Menger’s Theorem, and are connected by at least three pairwise internally disjoint paths. Suppose that , , and are three pairwise internally disjoint paths connecting and . We consider the graph induced by , , and . By Rayleigh’s short-cut principle, . Suppose that the lengths of , , and are , , and , respectively. Then by the series and parallel connection rules of resistors, Since it is obvious that , it follows that Thus .
By Foster’s third formula (30) and noticing that there exist 3-walks which are not 3-path in , we conclude that Hence
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this article.
The author would like to thank the anonymous referee for his/her careful reading of the paper and valuable comments and suggestions. The support of National Natural Science Foundation of China through Grants no. 11201404 and 11371307, China Postdoctoral Science Foundation through Grants no. 2012M521318 and 2013T60662, and Special Funds for Postdoctoral Innovative Projects of Shandong Province through Grant no. 201203056 is greatly acknowledged.
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