1. Introduction
Thus far, chemistry has been extensively studied and applied in graph theory [
1,
2,
3,
4]. It is often the case that chemical compounds can be interpreted by chemical graphs, in which vertices represent the atoms and edges stand for the covalent bonds between atoms. The random regular polygonal chains studied here are obtained by randomly connecting
n regular polygons through vertex-to-vertex in turn; these connection vertices are termed cut vertex. Meanwhile, the random regular polygonal chains with
n regular polygons are spiro compounds, which are a significant class of cycloalkanes in organic chemistry; refer to [
5,
6]. Spiro compounds have a rigid and stable structure, their chiral ligands have a large specific optical rotation, and they have important applications in asymmetric catalysis, luminescent materials, pesticides, polymer binders and so on.
In theoretical chemistry, topological indices are commonly used to encode molecules, as well as to design compounds with physicochemical properties or biological activity [
7]. It is a popular issue to study the topological indices of the chemical chains and their significance.
Throughout this paper, all the graphs that we take into account are finite, undirected and simple; a graph is simple if it has no loops and no two of its links join the same pair of vertices. More detailed notations and terminologies can be found in [
8]. Let
be a disordered graph;
represents the vertex set and
stands for the edge set. The degree
, or
for short, is the number of edges connected to the node
u(
). The length of the shortest path between vertex
u and vertex
v(
) is defined as the distance
(short for
). The sum of the distances between any two vertices in
is
.
The Wiener index [
9,
10,
11,
12,
13,
14] is the best-known of all topological indices, which is defined as
. By weighting the Wiener index, the Gutman index is proposed as
which is a kind of vertex-valency-weighted sum of the distance between all pairs of vertices in a graph [
15,
16,
17,
18,
19]. The index is closely related to the Wiener index and has the same structured features as the Wiener index in the condition of acyclic structures. Therefore, the research for the possible chemical applications of the Gutman index and the theoretical investigations should lay particular emphasis on the condition of polycyclic molecules.
The Schultz index [
20] is proposed as
The Schultz index has been shown to be a useful molecular descriptor in the design of molecules with desired properties [
21]; thus, further studies on the mathematical and computational properties of the Schultz index are desirable, including more articles on developing such topology indices of the mathematical properties and applications [
22].
Many scholars have studied the indices of some random polyphenylene chains before [
23,
24,
25,
26,
27,
28]. Recently, Liu and Zeng et al. [
29] obtained some indices in the random spiro chains, including the Gutman index, Schultz index, multiplicative-degree Kirchhoff index and additive-degree Kirchhoff index, and determined the expected values of these indices in the random spiro chain, and the extremal values among all spiro chain with
n hexagons. Motivated by [
29,
30,
31], we explore the property of the Gutman index and Schultz index of polygonal regular chains, and determine the expected values of the indices, denoted by
and
in the random regular polygonal chains with
n regular polygons, and we discuss the maximum and minimum of the
and
. Meanwhile, we acquire the extremal values and the average values of the Gutman index and Schultz index among all regular polygonal chains with
n regular polygons.
The random regular polygonal chain
with
regular polygons consists of a new regular polygon
attaching to the end of a regular polygonal chain
with
n regular polygons; see
Figure 1.
For
, there are
k ways to connect the terminal regular polygon
with the front random regular polygonal chain
. They can be described as
and
, respectively; see
Figure 2.
A random regular polygonal chain with n regular polygons can be obtained by adding a regular polygon at the end of the chain step by step. At each step , a random connection is made from one of the following k possible cases:
with probability ,
with probability ,
with probability ,
where the k probabilities and are independent of the step parameter t. When , the meta-chain can be obtained. When , the orth-chain can be obtained. When , the orth-chain can be obtained. When , the other probabilities reach 0, and we can obtain the orth-chain . When , the para-chain can be obtained.
2. The Gutman Index of the Random Regular Polygonal Chain
For the random regular polygonal chain
, we denote by
the graph acquired by connecting a new terminal regular polygon
to
, which is spanned by vertices
and
is
(see
Figure 1). It is evident that, for all
, we have
Meanwhile, we obtain that
Theorem 1. The of the random regular polygonal chain is Proof. Let
.
Then, .
For convenience, we could denote the random variable
by
Therefore, we can obtain a recurrence relation as follows:
By considering the following k possible conditions, it is easy to obtain .
Condition 1. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 2. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 3. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 4. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 5. , then would cover the vertex . Hence, can be represented as with probability .
According to the
k conditions, we have that
Moreover, the original value is
. Thus,
From the original value,
, and the above recurrence relation, we may calculate that
The proof is complete. □
According to the proof of Theorem 1, if , each of the other probabilities is equal to 0, and then we have . Likewise, if , each of the other probabilities is equal to 0, and then ; if , each of the other probabilities is equal to 0, and then and so on; if , each of the other probabilities is equal to 0, and then ; if , each of the other probabilities is equal to 0, and then .
Corollary 1. The Gutman indices of , , are Corollary 2. Among all polygonal chains with 2k polygons, realizes the maximum of and realizes the minimum of .
Proof. By Theorem 1, we have
as
, we have that
When
,
realizes the maximum of
, thus,
. If
,
, we can translate this into
, then
However,
(i.e.,
), and we cannot acquire the minimum value of
. Thus, we consider
, then
.
However,
(i.e.,
), and
cannot attain the minimum value. By this analogy, if
, let
.
Therefore, achieves the minimum value, if , which will be . This completes the proof. □
3. The Schultz Index of the Random Regular Polygonal Chain
The Schultz index of a random polygonal chain is a random variable. We calculate the expected value of as follows. Denote by the expected value of the Schultz index of the random polygonal chain .
Theorem 2. The of the random regular polygonal chain is Proof. Let
.
Then, .
For convenience, we could denote the random variable
by
Therefore, we can obtain a recurrence relation as follows:
By considering the following k possible conditions, it is easy to obtain .
Condition 6. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 7. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 8. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 9. , then would cover the vertex or . Hence, can be represented as or with probability .
Condition 10. , then would cover the vertex . Hence, can be represented as with probability .
According to the
k conditions, we have that
Moreover, the original value is
. Thus,
From the original value,
, and the above recurrence relation, we may calculate that
The proof is complete. □
According to the proof of Theorem 2, if , each of the other probabilities is equal to 0, and then we have . Likewise, if , each of the other probabilities is equal to 0, and then ; if , each of the other probabilities is equal to 0, and then and so on; if , each of the other probabilities is equal to 0, and then ; if , each of the other probabilities is equal to 0, and then .
Corollary 3. The Schultz indices of , , are Corollary 4. Among all polygonal chains with 2k polygons, realizes the maximum of and the meta-chain realizes that of the minimum.
Proof. By Theorem 2, we can deduce
as
, we have that
When
,
realizes the maximum of
; thus,
. If
,
, we can translate this into
, then
However,
(i.e.,
), and we cannot acquire the minimum value of
. Thus, consider
, then
.
However,
(i.e.,
), and
cannot attain the minimum value. By this analogy, if
, let
.
Therefore, achieves the minimum value, if , which will be . This completes the proof. □