Abstract
With generating function and structural analysis, this paper presents the subtree generating functions and the subtree number index of generalized book graphs, generalized fan graphs, and generalized wheel graphs, respectively. As an application, this paper also briefly studies the subtree number index and the asymptotic properties of the subtree densities in regular book graphs, regular fan graphs, and regular wheel graphs. The results provide the basis for studying novel structural properties of the graphs generated by generalized book graphs, fan graphs, and wheel graphs from the perspective of the subtree number index.
1. Introduction
The invariants of the graph and its topological indices have important significance in studying the properties of chemical molecular graphs and network graphs [1–4]. The Wiener index, for example, is defined as the sum of the distances of all disordered vertex pairs, and it can be used to estimate the boiling point of the alkanes [5, 6]. Therefore, new topological indices are constantly introduced by worldwide scholars in recent years.
As a representative structural based index, the subtree number index of a graph (also known as -index [7, 8]), which is defined as the number of nonempty labeled subtrees of , plays an important role in areas such as biological reconstruction [9], reliable network construction [10], and machine learning [11] and therefore has attracted much attention in recent years.
The subtree number problem was first proposed by Jamison, when studying the subtree average order of trees [12, 13]. Székely and Wang [8] characterized the extremal graph among all vertices tree. Meanwhile, they presented the subtree number formula of good binary tree and relation between the Wiener index and subtree number index of trees. Yan and Yeh [14] proposed a linear algorithm for enumerating subtrees of trees and further characterized the trees with the second to fifth largest numbers of subtrees and the tree with the second minimum number of subtrees through graph transformation and generating function. With generating function and cycle contraction carrying weights technique, Yang et al. [15–17] solved the subtree enumeration problem and characterized the extremal graphs among all hexagonal, phenylene chains, and spiro, polyphenyl hexagonal chains of length n, respectively.
Using the generating function method, more recently, Chin et al. [18] studied the enumeration problem of subtrees and spanning subtrees of complete graphs, complete bipartite graphs, and theta graphs. Yang et al. [17] studied the expected subtree number index in random polyphenylene and spiro chains. By introducing new “middle parts” of a tree, Li et al. [19] studied several extreme ratios of distance and subtree problems in binary trees. With the combinatorial technique, Kamiński and Prałat [20] provided the upper and lower bounds for the number of subtrees of random tree. Zhang et al. [21] presented the concepts of the eccentric subtree number and studied some extremal results with respect to this topological index in a tree.
Wheel graphs, fan graphs, and book graphs are three representative cyclic graphs. Various properties of these graphs have been studied, for example, Lin et al. [22] studied the Laplacian spectral features of vertex extension of wheel graphs and multifan graphs. Daoud [23] studied the number of spanning trees of wheel graphs and complex graphs generated by wheel graphs. Yang et al. [24] studied the subtree problem of wheel graphs and multifan graphs with the generating functions. Barioli [25] studied the complete matrix of book graphs. Based on wheel graph and chicken swarm optimization, Yu et al. [26] proposed a novel hybrid localization scheme for deep mine. Liu et al. [27] proved that multifan graphs were determined by their Laplacian spectra.
We first introduce related terminologies and notations in Section 2. The subtree generating functions and subtree number indices of generalized book graphs , fan graphs , and wheel graphs are derived in Section 3. Some special cases are also discussed as immediate consequences. In Section 4, we analyze the asymptotic characteristics of the subtree densities of the aforementioned generalized graphs. We summarize our work in Section 5.
2. Terminology and Notation
Let be a weighted graph with vertex set , edge set , vertex weight function , and edge weight function . Unless otherwise stated, we specify and , where is a commutative ring with a unit element 1. In what follows, we list the symbols that will be used in later discussion.(i)Denote by the leaf set of .(ii)Denote by the graph obtained from by removing from .(iii)Denote by the set of all subtrees of .(iv)Denote by a star with vertices and a path with vertices.(v)Denote by the set of subtrees of containing , where can be a vertex or an edge set.(vi)Denote by the set of subtrees of containing both vertex and path .
For any subtree , the weight of is defined asand the subtree generating function of is defined as . Similarly, we define , the subtree generating function of containing , and , the subtree generating function of containing both vertex and path .
Letting vertex and edge weight in the above generating functions, we have the corresponding subtree number indices, namely, , , and .
Next, we introduce some lemmas and definitions that will be used in our arguments. Let be a weighted tree on vertices, be a vertex of , be a pendant vertex, and be the pendant edge of . We define a weighted tree from as follows: , , andfor any , .
Lemma 1 (see [14]). With the above notations, we have
Definition 1. If the pendant edges of the star are replaced by paths of length , the resulting tree is called a generalized star tree or simply .
Definition 2. Define (or simply ) the -page generalized book graph constructed by joining the end vertices and of the internally disjoint paths () of length , where and . Obviously, has vertices and edges.
Definition 3. Define (or simply ) the generalized fan graph obtained by firstly constructing a generalized star through identifying the end vertices of the n internally disjoint paths together, where and then connecting vertex pair with path of length , where , , and (specifying that ). For brevity, we abbreviate some notations.(i)Denote , , .(ii)Denote , , , specifying that .
Definition 4. Define (or simply ) the generalized wheel graph constructed from the generalized fan graph F(n) (see Definition 3) by joining vertices pair and of the generalized fan graph with a path , where and . Similarly, we abbreviate some notations for the brevity.(i)Denote , , .(ii)Denote , , , , specifying that .
3. Subtree Generating Functions for Generalized Book Graphs, Fan Graphs, and Wheel Graphs
3.1. Subtree Generating Function for Generalized Book Graphs
Before deriving the subtree generating function for generalized book graphs, we first introduce some lemmas.
Lemma 2 (see [14]). Let be a weighted path on vertices, with , , and ; then,
Lemma 3 (see [15]). Let be a weighted graph with the common edge , , , , and , , where , are two weighted graphs; then,
Lemma 4 (see [16]). Let be a weighted unicyclic graph with , (where ), , and , respectively; then, for any fixed vertex and any continuous edge set , we have
The following Lemma follows from Lemmas 1 and 2.
Lemma 5. Let be a weighted generalized star tree, with and ; then,
Theorem 1. Let be an -page weighted generalized book graph, with and ; then,
Proof. We divide the subtrees of generalized book graph into four categories:(1)Containing but not .(2)Containing but not .(3)Containing neither nor .(4)Containing both and .With structural analysis and Lemma 5, we have the subtree generating function of cases (1) and (2) asWith Lemma 1, it is easy to see that the subtree generating function of case (3) isDenote by the path connecting the vertex pair , and an unicyclic graph. With structural analysis, equation (8) in Lemma 4, and Lemma 3, we have the subtree generating function of case (4) asThe theorem follows from equations (11)–(13).
We can further obtain the subtree generating function of special generalized book graphs.
Definition 5. Let be an -page generalized book graph and be a positive integer; we call an -page regular book graph, denoted by , if the lengths of its internally disjoint paths are all equal to (namely, ). Obviously, has vertices and edges.
With Theorem 1, it is not difficult to obtain the following corollary.
Corollary 1. Let be an -page regular book graph, with and ; then,
By substituting into equations (10) and (14), we have the corresponding subtree number indices.
Corollary 2. The subtree number index of the -page generalized book graph is
Corollary 3. The subtree number index of the -page regular book graph is
With Corollary 3, we have the subtree number index of the as illustrated in Table 1.
3.2. Subtree Generating Function of Two-Tailed Generalized Fan Graph
Before solving the subtree generating functions of generalized fan graphs and generalized wheel graphs, we firstly introduce and solve the subtree generating function computing problem of the two-tailed generalized fan graphs. Firstly, we employ the convention that ().
Definition 6. Denote by the generalized fan graph of that contains and ; moreover, define the path tree of that contains vertex .
With structural analysis and equations (9) and (13), we can obtain the following lemma.
Lemma 6. Keeping above symbols and definitions, we have
Definition 7. Define (or simply ) the two-tailed generalized fan graph obtained from by attaching a path to vertex and path , to vertex cn, respectively, where and (see Figure 1 for an illustration).(i)Denote by the regular two-tailed generalized fan graph with .(ii)Denote by and the two weighted trees.
(a)
(b)
(c)
(d)
Lemma 7. Let be a weighted two-tailed generalized fan graph, with and , respectively; then,with as in equation (17) and .
Proof. We categorize the subtrees of containing into four cases:where(i) is the set of subtrees in that does not contain edge .(ii) is the set of subtrees in that contains edge set , but not edge .(iii) is the set of subtrees in that contains edge set , but not edge .(iv) is the set of subtrees in that contains edge set .For the case , we define path tree , and it is not difficult to obtainFor the case , with structural analysis and equation (17), we obtainSimilar to analysis of case , we can obtainFor the case , with equations (5) and (17), we can getClearly, is a trivial tree, and by Lemma 1, we can getCombining equations (20)–(24), we can obtain equation (18), and thus the lemma is proved.
With Lemmas 1 and 7, we can obtain the subtree generating function of generalized fan graphs.
Theorem 2. Let be a weighted generalized fan graph, with and ; then,with as in equation (18).and , .
Next, we discuss some special cases of the generalized fan graphs.
Definition 8. Let be a generalized fan graph and be a positive integer; if , then the generalized fan graph is called the regular fan graph, denoted by . Obviously, has vertices and edges.
From Theorem 2, the following corollaries follow as immediate consequences.
Corollary 4. Let be a weighted regular fan graph, with and ; then,withand .and , .
By substituting into equations (25) and (27), we have the corresponding subtree number indices.
Corollary 5. The subtree number index of the generalized fan graph iswithand , .and .
Corollary 6. The subtree number index of the regular fan graph iswithand .and .
By Corollary 6, we can get the number of subtrees in as shown in Table 2.
3.3. Subtree Generating Functions of Unicyclic Graphs
In order to solve the subtree generating function of the generalized wheel graphs, we first study the subtree generating function of the unicyclic graphs.
Lemma 8 (see [14]). Let be a weighted tree with more than 2 vertices, and be its two distinct vertices, and be a path of connecting and with length , vertex set , and ; moreover, denote by the weighted tree of that contains , respectively. Then,where .
Theorem 3. Let be a weighted unicyclic graph with girth , vertex set , vertex weight function , and edge weight function , having trees attached to vertex on the cycle; then,where is the path connecting and ; here we specify that .
Proof. The theorem holds through dividing the subtrees of into the following three categories:(1)Containing no vertex and no edge on the cycle.(2)Containing only one vertex and no edge on the cycle.(3)Containing only vertices and edges on the cycle.
3.4. Subtree Generating Function of the Generalized Wheel Graphs
Next, we consider the subtree generating function of the generalized wheel graph .
Definition 9. Let be the generalized fan graph of that contains , and denote by the path tree of that contains .
With Lemma 6, the following lemma follows immediately.
Lemma 9. Keeping above symbols and definitions, we havewith
Theorem 4. Let be a weighted generalized wheel graph with vertex weight function and edge weight function (see Definition 4); then,with , , as in equations (18), (26), and (38), respectively, and .
Proof. We first divide the subtrees of generalized wheel graph into two categories:(1)Not containing the center vertex .(2)Containing the center vertex .Obviously, is a unicyclic graph; with Theorem 3, we have the subtree generating function of case (1) asWe further partition the subtree of case (2) into four categories.where(i) is the set of subtrees in that does not contain edge .(ii) is the set of subtrees in that contains edge set , but not .(iii) is the set of subtrees in that contains edge set , but not .(iv) is the set of subtrees in that contains edge set , but not . With equation (18), we have the subtree generating function of case as We further categorize the subtrees in set into two cases:(v) is the set of subtrees in that contains edge set , but not .(vi) is the set of subtrees in that contains edge set .Then, we haveSimilar to the analysis of case , we haveEquation (40) follows from equations (41) to (47).
With Theorem 4, we can also obtain the subtree generating functions of some special generalized wheel graphs.
Definition 10. Let be a generalized wheel graph and be a positive integer; the generalized wheel graph is called regular wheel graph, denoted by , if . Clearly, has vertices and edges.
Corollary 7. Let be a weighted regular wheel graph, with and ; then,with and as in equations (28) and (29), respectively.
Substituting into equations (40) and (48), we have the following corollaries.
Corollary 8. The subtree number index of generalized wheel graph iswith , as in equations (31) and (32), respectively, and by substituting into equation (38).
Corollary 9. The subtree number index of the regular wheel graph iswith and as in equations (34) and (35), respectively.
With Corollary 9, we can get the subtree number index of the regular wheel graph as shown in Table 3.
4. Subtree Density for Graphs
The subtree generating function can be used to reveal more structure information of the graph than just subtree numbers. Next, we study the asymptotic characteristics of the subtree density of three types of graphs, namely, regular book graph , regular fan graph , and regular wheel graph . The formal definition of the subtree density is given below.
Definition 11. (see [12]). Assume is a graph with vertices and subtrees of order . Then, denotes the average order of subtrees of , and the subtree density of is defined as .
By Definition 11, the subtree density of is simplywhere represents the vertex generating function of , represents the number of subtrees of , and represents the number of vertices of .
4.1. Subtree Density of the Regular Book Graph
According to definition of regular book graph, it is easy to know that the number of vertices in is
With Corollary 1, we obtain the vertex generating function of :
Together with equations (16) and (51), we can obtain the subtree densities of as plotted in Figure 2, and the related data are listed in Table 4.
Observing Table 4 and Figure 2, we see that the has the maximal subtree density among . On the whole, it appears that the subtree density of decreases with the increase of and .
4.2. Subtree Density of the Regular Fan Graph
Similarly, with Corollary 4, we can obtain the vertex generation function of , and combining the subtree density definition (see equation (51)), we can obtain the subtree densities of as shown in Table 5 and Figures 3(a) and 3(b).
(a)
(b)
From Figure 3(a), when is fixed, we can observe that the subtree density of gradually increases with the increase of . As illustrated in Figure 3(b), when is fixed, the subtree density of also increases with the increase of .
4.3. Subtree Density of the Regular Wheel Graph
With Corollary 7 and equations (50) and (51), we can also obtain the subtree density of as shown in Table 6 and Figure 4; here, we skip the repetitive analysis details.
(a)
(b)
From Figure 4(a), when is fixed, we notice a very interesting observation: the subtree density of gradually decreases with the increase of . This trend needs to be further verified through combinatoric approaches. When is fixed, by observing Figure 4(b), it appears that the subtree density of increases with the increase of .
5. Conclusion
As an important topological index of graphs and networks, the subtree number index is closely related to the analysis of the hybrid local reliable network and discovery of new properties of compounds. We present the subtree generating functions and subtree number indices of the generalized book graph , generalized fan graph , and generalized wheel graph . Furthermore, we also briefly discuss the subtree density asymptotic characteristics of the regular cases of these three graphs: , , and . Our study provides some basis for exploring hidden properties of complex cyclic graphs and chemical compounds from a new metric.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (grant nos. 61702291 and 11801371), the Program for Science & Technology Innovation Talents in Universities of Henan Province (grant no. 19HASTIT029), the Key Research Project in Universities of Henan Province (grant nos. 19B110011 and 19B630015), and the Scientific Research Starting Foundation for High-Level Talents of Pingdingshan University (grant no. PXY-BSQD2017006).