Let be a graph with vertex set . Let be the degree of the vertex . If the vertices form a path of length in the graph , then the th order Randiฤ index of is defined as the sum of the terms over all paths of length contained (as subgraphs) in . Lower and upper bounds for , in terms of the vertex degree sequence of its factors, are obtained for corona product graphs. Moreover, closed formulas are obtained when the factors are regular graphs.
1. Introduction
In this work we consider simple graphs with vertices and edges. Let be the vertex set of . For every vertex , represents the degree of the vertex in . The maximum and minimum degree of the vertices of will be denoted by and , respectively.
The Randiฤ index of a graph was introduced in 1975 [1] and defined as
This graph invariant, sometimes referred to as connectivity index, has been successfully related to a variety of physical, chemical, and pharmacological properties of organic molecules, and it has became into one of the most popular molecular-structure descriptors. After the publication of the first paper [1], mathematical properties of were extensively studied, see [2โ6] and the references cited therein.
The higher-order Randiฤ indices are also of interest in chemical graph theory. For , the th order Randiฤ index of a graph is defined as
where denotes the set of paths of length contained (as subgraphs) in . Of the higher-order Randiฤ indices the most frequently applied is [7โ10]. Estimations of the higher-order Randiฤ index of regular graphs and semiregular bipartite graphs are given in [10]. In this paper we are interested in studying the higher-order Randiฤ index, , for corona product graphs. Roughly speaking, we study the cases , for arbitrary graphs and the case when the second factor of the corona product is an empty graph. As an example of a chemical compound whose graph is obtained as a corona product graph we consider the Cycloalkanes with a single ring, whose chemical formula is , and whose molecular graph can be expressed as , where is the cycle graph of order and is the empty graph of order two. We recall that, given two graphs and of order and , respectively, the corona product is defined as the graph obtained from and by taking one copy of and copies of and then joining by an edge each vertex of the copy of with the vertex of .
2. Estimating for Corona Graphs
Theorem 2.1. For , let be a graph of minimum degree , maximum degree , order and size . Then,
Proof. Let , , and let . We have
where
Thus, the lower bound follows. Analogously we deduce the upper bound.
Corollary 2.2. For , let be a -regular graph of order . Then,
Theorem 2.3. For , let be a graph of minimum degree , maximum degree , order , and size . Then,
Proof. Let and be the set of vertices of and , respectively. Given a vertex , we denote by the set of neighbors that has in . The paths of length two in are obtained as follows: (i)paths , , where and , (ii)paths , , where and , (iii)paths , , where and , (iv)paths of length two belonging to , (v)paths of length two belonging to the copies of .
So, we have , where
corresponds to the paths type (i),
corresponds to the paths type (ii),
corresponds to the paths type (iii),
corresponds to the paths type (iv), and
corresponds to the paths type (v). Thus, the lower bound follows. The upper bound is obtained by analogy.
Corollary 2.4. For , let be a -regular graph of order . Then,
The girth of a graph is the size of its smallest cycle. For instance, the molecular graphs of benzenoid hydrocarbons have girth 6. The molecular graphs of biphenylene and azulene have girth 4 and 5, respectively [11].
The following result, and its proof, was implicitly obtained in the proof of Theorem 1 of [10]. By completeness, here we present it as a separate result.
Lemma 2.5. Let be a graph with girth . If and , then the number of paths of length in is bounded by
Proof. Since , for every , the number of paths of length 2 in of the form is . Therefore, the result follows for . Suppose now that . Given a vertex , let be the set of paths of length whose second vertex is , that is, paths of the form . We denote by the set of neighbors of an arbitrary vertex . Note that the degree of is . If , then for every and we have . So, for every , there exists a vertex sequence such that , , . If , then the sequence is a path. Conversely, every path of length whose second vertex is can be constructed as above. Hence, the number of paths of length whose second vertex is is bounded by
Thus, the result follows.
Now denotes the empty graph of order .
Theorem 2.6. Let be a graph with girth , minimum degree , and maximum degree . If and , then
Proof. The paths of length in contribute to in
Moreover, each path of length in leads to paths of length in ; thus, the paths of length in contribute to in
Hence,
By Lemma 2.5 we obtain the upper bound and the lower bound is obtained by analogy.
Corollary 2.7. Let be a -regular graph of order and girth . If and , then
Acknowledgment
This work was partly supported by the Spanish Government through projects TSI2007-65406-C03-01 โE-AEGISโ and CONSOLIDER INGENIO 2010 CSD2007-00004 โARES.โ
References
M. Randiฤ, โOn characterization of molecular branching,โ Journal of the American Chemical Society, vol. 97, no. 23, pp. 6609โ6615, 1975.
Y. Hu, X. Li, Y. Shi, T. Xu, and I. Gutman, โOn molecular graphs with smallest and greatest zeroth-order general Randiฤ index,โ Match, vol. 54, no. 2, pp. 425โ434, 2005.
O. Araujo and J. A. De La Peรฑa, โThe connectivity index of a weighted graph,โ Linear Algebra and Its Applications, vol. 283, no. 1โ3, pp. 171โ177, 1998.
I. G. Yero, J. A. Rodrรญguez-Velรกzquez, and I. Gutman, โEstimating the higher-order Randiฤ index,โ Chemical Physics Letters, vol. 489, pp. 118โ120, 2010.
