1. Introduction
In this paper we deal with finite undirected connected graphs
G with vertex set
. The
degree of a vertex
v is denoted by
. If
u and
v are vertices of
G, then the
distance is the number of edges in a shortest path connecting them. For a vertex
, define its distance as
. The
Wiener index is a distance-based topological index of molecular graphs. It is defined as the sum of distances between all unordered pairs of vertices in
G:
It was introduced as a structural descriptor for characterization of acyclic structures [
1]. The Wiener index and its numerous modifications are intensively studied in mathematical and theoretical chemistry and have found various applications in the modeling of physico-chemical, biological, and pharmacological properties of organic molecules. Mathematical properties and chemical applications of the Wiener index can be found in numerous books and reviews (see selected books [
2,
3,
4,
5,
6,
7,
8] and articles [
9,
10,
11,
12,
13,
14]). One of the directions of studying of the Wiener index is the development of methods for calculating the index for composite graphs (see [
13,
14,
15,
16,
17,
18,
19,
20,
21]). Since chemical reactions lead to transformations of the structure of molecular graphs, these methods are useful for the evaluation the change in the Wiener index during molecular rearrangements of different types. In general case, the Wiener index of a composite graph depends on Wiener indices of the initial graphs and distance properties of the identified elements in these graphs. An interesting kind of composite graphs is so-called thorny graphs. The concept of thorny graphs was introduced by Gutman in [
22]. The thorny graph of
G is the graph obtained by attaching pendant vertices to all vertices of
G. The Wiener index of the thorny graph does not depend on distance properties of vertices of
G in some cases. Eventually, this concept found a variety of chemical applications [
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33].
In this paper, we consider the Wiener index of a new kind of thorny-like graphs. An edge thorny graph G of a benzenoid graph H is constructed by attaching new graphs to edges of H. The specific property of thorny graphs is preserved: the index of the resulting graph does not contain distance characteristics of vertices of H and depends on the Wiener index of H and distance properties of the attached graphs.
2. Catacondensed Benzenoids Graphs
Benzenoid graphs are composed of six-membered cycles (hexagonal
rings) connected to each other along the edge. We assume that such a graph contains at least two hexagonal rings. Any two rings either have one common edge (and are then said to be adjacent) or have no common vertices, and no three rings share a common vertex. Each hexagonal ring is adjacent to two or three other rings, with the exception of the
terminal rings to which a single ring is adjacent. The
inner dual graph of a given benzenoid graph consists of vertices corresponding to hexagonal rings of the graph; two vertices are adjacent if and only if the corresponding rings share an edge. A benzenoid graph is called
catacondensed if its inner dual graph is a tree. Denote by
the set of all catacondensed benzenoid graphs with
h rings. Set
include molecular graphs of catacondensed benzenoid hydrocarbons [
6]. Benzenoid hydrocarbons are important raw-materials of the chemical industry (used, for example, for production of dyes and plastics), but are also dangerous pollutants. Since this class of chemical compounds is attracting much attention from theoretical chemists, the theory of the Wiener index of the respective molecular graphs has been developed for many years [
34,
35].
Linear benzenoid graph contains
h linearly connected rings. Catacondensed benzenoid graphs without subgraph
form subset
. Chains with this property are known under the name fibonacenes [
36]. The Wiener index of fibonacenes was studied in [
17,
37,
38,
39,
40,
41]. Examples of
Figure 1 are the linear benzenoid graph
, a fibonacene of
and a branched benzenoid graph of
.
A benzenoid graph with
h hexagonal rings has
vertices and every vertex has degree 2 or 3. Edges with incident vertices of the same degree will be called 2-edges and 3-edges. Incident vertices of 2,3-edges have degree 2 and 3. The vertex set of a graph
H can be split into two disjoint subsets with respect to vertex degrees:
, where vertex subsets
and
have cardinality
and
[
35]. Since a benzenoid graph is bipartite,
.
3. Structure of Attached Graphs
In this section, we describe a way of attaching of graphs
F,
E, and
T to edges of a benzenoid graph
H. An attachment scheme of the graphs is depicted in
Figure 2. Edge
of the graphs is identified with 2-edges, 3-edges, and 2,3-edges of
H as shown in
Figure 2. Vertex
v of
T should be always identified with a vertex of degree 2 of
H.
We will assume that the attached edge of F, E, and T does not belong to a cycle with an odd number of vertices. Then we can define the subgraphs induced by vertex sets containing vertices v and u. Namely, , , , , , and . The structure of these subgraphs may be different with one exception: and . Therefore, , , and .
A
perfect matching of a graph is a set of mutually nonadjacent edges that spans all vertices of the graph. Perfect matchings play an important role in the studies of benzenoid hydrocarbons [
6,
42]. A graph
G is called the
edge thorny graph if it is constructed by joining new graphs to all edges of a perfect matching in a catacondensed benzenoid graph
H. Each vertex of
H is covered by only one attached graph.
Denote by and the path and the cycle with n vertices. Equalities or imply that or . Attaching such graphs does not change the structure of a benzenoid graph. Equalities and imply . In this case, a benzenoid graph and its edge thorny graph coincide. If edge is a bridge in F, E, and T, then the edge thorny graph may be regarded as a result of joining subgraphs , , , , , and to single vertices of a benzenoid graph.
4. Main Result
Let M be a perfect matching of catacondensed benzenoid graph and M has a 2-edges, b 3-edges, and c 2,3-edges, i.e., . Let graph G be obtained from H by attaching graphs F, E and T to all edges of M as described in the previous section. Then the Wiener index of G does not depend on distance properties of vertices and edges of H.
Theorem 1. If G is the edge thorny graph of a benzenoid graph , thenwhere coefficients are independent of the distances in the graphs: , , , and , where . As an illustration, consider the calculation of the Wiener index of an edge thorny graph
G for benzenoid graph
shown in
Figure 3. Wiener indices of graphs of the figure are
,
,
, and
. Distances of vertices are
,
,
,
, and
. Graph
F is attached to
edges and graph
E is attached to
edges. By Theorem 1,
Theorem 1 has been proved by a long combinatorial reasoning in
Section 6. Although the obtained formula is quite cumbersome, in some cases it leads to simple expressions for the Wiener index.
If then the edge thorny graph coincide with the initial benzenoid graph. The next two corollaries immediately follow from Theorem 1. The Wiener index of an edge thorny graph does not change if vertices v and u of graphs F and E are swapped.
Corollary 1. Let graph be obtained from an edge thorny graph by swapping vertices v and u of a copy of graph F or E. Then .
Two perfect matchings are called equivalent if they contain the same number of 2-edges, 3-edges, and 2,3-edges.
Corollary 2. Let edge thorny graphs and be obtained from a benzenoid graph H by attaching graphs F, E, and T to the edges of equivalent perfect matchings of H. Then .
If all attached graphs have the same parameters involved in Theorem 1 (say, they are isomorphic), then the formula for the Wiener index has a simple form.
Corollary 3. Let G be the edge thorny graph of a benzenoid graph . If and , theni.e., the Wiener index of G does not depend on the choice of a perfect matching in H. For example, if a new hexagonal ring is attached to all edges of an arbitrary perfect matching of
, then
Specifying the structure of graphs F, E, and T, one can construct edge thorny graphs relevant to chemical graphs.
5. Examples of application of Theorem 1
Consider a perfect matching of H without 2,3-edges. Let . Since 2-edges should cover all vertices of degree 2, suitable catacondensed benzenoid graphs belong to . This implies that and , , , and . Then the formula of Theorem 1 is reduced to the following expression.
Corollary 4. If G is the edge thorny graph of a benzenoid graph , and , then An example of the edge thorny graph
G of
is shown in
Figure 4. The numbers of 2-edges and 3-edges of the selected perfect matching are
and
, respectively. The other parameters of graphs are
,
,
,
, and
. Then
If
, then the edge thorny graph is also a benzenoid graph. Let
and a perfect matching of
contains
a 2-edges and
3-edges. Denote by
the edge thorny graph of benzenoid graph
,
. It is clear that
for some
. Applying Corollary 4, we get the following recurrent relation
It is easy to see that
and
,
. Solving relation (
1), we obtain
Apply this formula for calculation the Wiener index of graphs
shown in
Figure 5.
Inner dual graphs of
,
, may be regarded as a growing tree-like dendrimer with non-pendant vertices of degree 3. Since
,
and
, we obtain
The Wiener index of the initial edge thorny graphs are , , and .
If edge
of
F is a bridge, then the edge thorny graph can be considered as the result of attaching graphs
and
to single vertices of a benzenoid graph. Note that if subgraphs
and
of an edge are swapped, then the Wiener index will not change (see Corollary 1). For instance, edge thorny graphs
G and
of
Figure 6 are obtained from benzenoid graph
. Graphs
and
are isomorphic to
,
,
,
,
, and
. By Corollary 4, we have
and
Consider the case when graph T is also used for constructing edge thorny graphs.
Corollary 5. Let G be the edge thorny graph of and . Thenwhere , , and . Consider an example of the edge thorny graph
G of a benzenoid graph
depicted in
Figure 7. A selected perfect matching of
H has
2-edges,
3-edge, and 4 2,3-edges. Parameters of graphs of
Figure 7 are
,
,
,
,
,
,
,
, and
. Then
The next example illustrates the estimation of the change in the Wiener index for two structures with the same number of vertices after removing some edges. Let graphs
and
be obtained from the benzenoid graph
H with four hexagonal rings by attaching graphs
and paths
to
edges of
H as shown in
Figure 8 (3-edges of
H are covered by graphs
). By Corollary 4, one can easily calculate that
6. Proof of Theorem 1
Let graph G be obtained from an n-vertex benzenoid graph by identifying edges of a perfect matching of H with edges of graphs F, E, and T. Denote by , , and copies of graphs F, E, and T attached to a 2-edges, b 3-edges, and c 2,3-edges of the perfect matching, respectively. The distance between two vertex subsets X and Y of a graph is defined as .
Consider the sums of vertex distances with respect to vertex degrees. Define the following parts of the Wiener index of
:
and
, where
. Since each distance is counted twice in
,
,
, and
,
The following decomposition lemma of the Wiener index will be used in the proof of Theorem 1.
Lemma 1. [
43]
For a catacondensed benzenoid graph H with h rings, , , , , and . Let
and
. Then the Wiener index of the edge thorny graph
G can be represented as follows.
Next, we find summands of Equation (
2) for graphs
H and
T. Calculations for graphs
F and
E are similar.
1. Let
. Then
Summing the last equation for all
, we have
2. Let
. Then
Summing the last equation for all
, we get
For all graphs
, we obtain
After similar calculations for graphs
, we have
Now we are ready to write the final contribution of graphs
to
:
To find contributions of graphs
F and
E, we have to do the same calculations. Then
and
It is not hard to derive that for graph
,
After substitution expressions (
3), (
4)–(
6), and (
7) back into Equation (
2), we have
In order to complete the proof, we need express quantities , , , , and in terms of (see Lemma 1) and rewrite in terms of and .
The proof is complete.
7. Conclusions
The Wiener index of edge thorny graphs of benzenoid graphs is studied. An edge thorny graph is a kind of composite graph. It is obtained by attaching new graphs to the edges of an original benzenoid graph H. These edges cover all the vertices of H and, moreover, form a perfect matching in H. It is shown that the index of the resulting graph does not contain distance characteristics of vertices or edges of H and depends on the Wiener index of H and distance properties of the attached graphs. The obtained formulas may be useful for calculation the Wiener index of some classes of chemical graphs. In particular, by specifying the attached graphs, one can evaluate the change in the Wiener index for molecular graphs under structural rearrangements.