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Hosoya properties of the commuting graph associated with the group of symmetries

  • Ghulam Abbas , Anam Rani , Muhammad Salman , Tahira Noreen and Usman Ali EMAIL logo
Published/Copyright: June 18, 2021
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Main Group Metal Chemistry
From the journal Main Group Metal Chemistry Volume 44 Issue 1

Abstract

A vast amount of information about distance based graph invariants is contained in the Hosoya polynomial. Such an information is helpful to determine well-known distance based molecular descriptors. The Hosoya index or Z-index of a graph G is the total number of its matching. The Hosoya index is a prominent example of topological indices, which are of great interest in combinatorial chemistry, and later on it applies to address several chemical properties in molecular structures. In this article, we investigate Hosoya properties (Hosoya polynomial, reciprocal Hosoya polynomial and Hosoya index) of the commuting graph associated with an algebraic structure developed by the symmetries of regular molecular gones (constructed by atoms with regular atomic-bonding).

Keywords: geodesic; Hosoya polynomial; reciprocal Hosoya polynomial; Hosoya index; molecular struture; network
MSC 2010: 15A27; 05C07; 05C12; 05C92

1 Introduction

A numeral quantity which captures the symmetry of a molecular structure is known as a topological index. In fact, a topological index is a numerical characterization of a chemical graph and it provides a mathematical function of the structure in quantitative structure-activity relationship (QSAR)/quantitative structure-property relationship (QSPR) studies. It correlates certain physico chemical properties such as boiling point, stability and strain energy of chemical compounds of a molecular structure (graph). Several properties of chemical compounds in a molecular structure can be determined with the assistance of mathematical languages rendered by various types of topological indices. Harold Wiener, introduced the concept of first (distance based) topological index while working on the boiling point of Paraffin and named this index as path number (Wiener, 1947). Next off, it was renamed as the Wiener index, and that was the time theory of topological indices began.

We consider simple and connected graph (chemical structure) G with vertex set V(G) and edge set E(G). We denote the two adjacent vertices u and v in G as u ~ v and non-adjacent vertices as u ≁ v. Starting from a vertex u and ending at a vertex v in G, a shortest alternating sequences of vertices and edges without repetition of any vertex is known as a u − v geodesic. The number of edges in a u − v geodesic is denoted by d(u,v), and is called the distance between u and v in G. The sum of two graphs G1 and G2, denoted by G1 + G2, is a graph with vertex set V(G1)∪V(G2) and an edges et E(G1)∪E(G2)∪ {u ~ v : u ∈V(G1) ∧ v ∈ V(G2)}. The maximum distance of a vertex v among all of its distances with all the vertices of G is called the eccentricity of v, denoted by ecc (v). The number diam(G)=maxv∈V(G)ecc(v) is the diameter of G.

1.1 Hosoya properties

Many chemist used the concept of counting polynomial, introduced by Polya (1936), in order to obtain the molecular orbitals of unsaturated hydrocarbons. In this regard, the spectra of the characteristic polynomial of graphs were studied extensively. In 1988, Hosoya used this concept to find the polynomials of many chemical structures (Hosoya, 1986b), known as Hosoya polynomials and seeks a lot of attention afterwards. In 1996, the Hosoya polynomial was renamed as Wiener polynomial by Sagan et al. (1996), but now a days, majority of researchers used the term Hosoya polynomial. The Hosoya polynomial provides a pile of information about distance based graph invariants. The relationship between the Hosoya polynomial and the hyper Wiener index was observed by Cash (2002). Several other applications of extended Wiener indices were studied by Estrada et al. (1998).

Consider a connected graph G of n vertices. Hosoya defined the polynomial of G with variable x in the following way:

H(G,x)=∑k≥0d(G,k)xk.

The coefficient d(G,k) denotes the number of unordered pairs (u,v) of vertices such that d(u,v) = k, where k ≤ diam(G). The reciprocal status Hosoya polynomial of G was introduced by Ramane and Talwar (2019), and is defined as follows:

Hrs(G,x)=∑u~v∈E(G)xrs(u)+rs(v),

where:

rs(v)=∑u∈V(G),v≠u1d(v,u)

is called the reciprocal status (transmission) of a vertex v.

The Hosoya index or Z-index of a graph G is the total number of its matchings (independent edge subsets, the empty set is also consider one matching in G). The Hosoya index is the prominent example of topological indices which is of great interest in combinatorial chemistry. The Z-index was introduced by Hosoya (1971, 1986a) and it turned out to be applicable to molecular chemistry, such as boiling point, entropy or heat of vaporization are well studied. Several researcher studied extremal problems with respect to the Hosoya index for various graph structures. With respect to Hosoya and Merrifield-Simmons indices, extremal properties of various graph, trees and unicyclic graph have been studies in the articles of Deng and Chen (2008), Wagner (2007), and Yu and Tian (2006). The minimal Hosoya index for cyclic systems was computed by Hou (2002). In 2008, Deng investigated the largest Hosoya index for every simple connected garph having n vertices and n +1 edges ( Deng, 2008).

1.2 Group of symmetries and commuting graph

Group of symmetries finds its notable use in the theory of molecular vibrations and electron structures. Due to their noteworthy employment in chemical structures, in the context of topological indices, we consider the group of symmetries of a regular molecular polygon (also called a regular molecular n-gon constructed on n ≥ 2 atoms with their regular atomic-bonding). A regular molecular n-gon is a molecular structure whose corners are atoms and sides are atom-bonds of same length, and each internal angle between atom-bonds is of the measurement π−2kπn radian. The group of symmetries of a regular molecular n-gon consists of 2n elements, which are n rotations about its center through an angle of 2kπn radian, where k = 0,1,…,n −1, either all clockwise or all anti-clockwise) and n reflections (for even n, the reflections through a line joining the mid-points of the opposite atom-bonds or through a line joining two opposite atoms; and for odd n, the reflections through those lines which join an atom with the mid-point of the opposite atom-bond). Symbolically, the group of symmetries is denoted by Dn and is called the dihedral group (a group theoretic name) of order 2n ( Majeed, 2013). If we denote a rotation by ‘a’ and a reflection by ‘b’, then 2n elements of Dn are a,a2 ,⋯,an−1 ,an = e and b,ab,a2b,⋯,an−1b (here e is the identical shape molecular n – gone). In generating form, Dn can be represented as follows:

  • Dn=〈a,b|an=b2=e,ab=ba−1〉.

    The center of Dn is:

    ζ(Dn)={{e},when n is odd,{e,an2},when n is even.

    Let Ω1 = {e,a,a2, ⋯,an−1}, Ω2 = {b,ab,a2b,⋯,an−1b} and Ω3 = Ω1 − ζ (Dn). Then |Ω1| = n = |Ω2| and

    |Ω3|={n−1,when n is odd,n−2,when n is even.

    In the case of even value of n ≥ 4, we partitioned Ω2 into n2 two element subsets Ω2i={aib,an2+ib} , 0≤i≤n2−1 , so that Ω2=∪i=0n2−1Ω2i .

    The commuting graph of a non-abelian group Γ is denoted by ΓG =C(Γ, Ω) with vertex set Ω ⊆ Γ. For two unlike elements x,y ∈ Ω,x ~ y in ΓG if and only if xy = yx in Γ. The notion of commuting graphs on non-central elements of a group has been studied by many researchers – see for instance Ali et al. (2016) and Bunday (2006), and the references therein. The commuting graph on Dn is defined by Ali et al. (2016) in the following result.

Proposition 1

For all n ≥ 3, let ΓG = 𝒞(Dn, Dn) be a commuting graph on Γ = Dn, then:

ΓG={K1+(K|Ω3|∪N|Ω2|),when n is odd,K2+(K|Ω3|∪n2K2),when n is even.

Here K1 is the trivial graph, Kp is a complete graph on p vertices, Nt is a null graph on t vertices, n2K2 is the union of n2 copies of K2.

2 Hosoya polynomials

In this section, we find Hosoya polynomial and reciprocal status Hosoya polynomial of ΓG.

2.1 Hosoya polynomial

Our first two results of this section provide the coefficients for the Hosoya polynomial of the commuting graph on Dn.

Proposition 2

Consider the commuting graph ΓG associated with the group Γ = Dn for odd values of n ≥ 3. Then:

d(ΓG,k)={2n,for k=0,n(n+1)2,for k=1,3n(n+1)2,for k=2.

Proof

Note that diam(ΓG) = 2. So, we have to find d(ΓG,0), d(ΓG,1) and d(ΓG,2). Consider the set Vp of all the pairs of vertices (same and distinct) of ΓG, then:

|Vp|=(|ΓG|2)+|ΓG|=n(2n+1).

Let S(ΓG,k) ={(l, m);l, m ∈V(ΓG)|d (l, m) = k} and d(ΓG,k) = |S(ΓG,k)|. Then:

(1) Vp=S(ΓG,0)∪S(ΓG,1)∪S(ΓG,2).

Since, d(l, l) = 0 for all l∈V(ΓG), so S (ΓG,0) = {(l, l);l ∈V(ΓG)}=V(ΓG). Thus, d(ΓG,0) = 2n. By Proposition 1, ΓG =K1+(K|Ω3| ∪ N|Ω2|) with V(K1) =ζ(Γ), V(K|Ω3|) = Ω3 and V(N|Ω2|) = Ω2. Therefore:

S(ΓG,1)={(l,m);l∈ζ(Γ),m∈Ω3}∪{(l,m);l∈ζ(Γ),m∈Ω2}∪{(l,m);l,m∈Ω3 and l≠ m}.

Accordingly,

d(ΓG,1)=n−1+n+(n−12)=n(n+1)2.

By Eq. 1, we have |Vp|=d (ΓG,0) +d (ΓG,1) +d(ΓG,2). Therefore:

d(ΓG,2)=|Vp|−d(ΓG,0)−d(ΓG,1)=n(2n+1)−2n−n(n+1)2=3n(n−1)2.

Proposition 3

Consider the commuting graph ΓG associated with the group Γ = Dn for even values of n ≥ 4. Then:

d(ΓG,k)={2n,for k=0,n(n+4)2,for k=1,3n(n−2)2,for k=2.

Proof

As diam(ΓG) = 2, so we have to find the coefficients d(ΓG,0), d(ΓG,1) and d(ΓG,2). If Vp denotes the set of all the pairs of vertices (same and distinct) of ΓG, then |Vp|=n(2n +1), by Proposition 2. Let S(ΓG,k) ={(l, m);l, m ∈V(ΓG)|d(l, m) = k}, then d(ΓG,k) =|S(ΓG,k)| and

(2) Vp=S(ΓG,0)∪S(ΓG,1)∪S(ΓG,2).

Since, d(l, l) = 0 ∀ l ∈ V(ΓG), so S(ΓG,0) ={(l, l); l ∈V(ΓG)} =V(ΓG). Thus, d(ΓG,0) = 2n. By Proposition 1, ΓG=K2+(K|Ω3|∪n2K2) with V(K2) =ζ(Γ), V (K|Ω3|) = Ω3 and V(n2K2)=∪i=0n2−1Ω2i=Ω2 . Thus:

S(ΓG,1)={(l,m);l∈ζ(Γ),m∈Ω3   ∪{(l,m);l∈ζ(Γ),m∈Ω2}∪{(l,m);(l,m)∈ζ(Γ) and l≠m}∪{(l,m);(l,m)∈Ω3 and l≠m}∪∪i=0n2−1{(l,m);(l,m)∈Ω2i and l≠m}.

Accordingly,

d(ΓG,1)=2(n−2)+2(n)+1+(n−22)+n2(1)=2n−4+5n2+1+(n−2)(n−3)2=n(n+4)2.

From Eq. 2, we have |Vp| =d(ΓG,0) +d(ΓG,1) +d(ΓG,2). Therefore

d(ΓG,2)=|Vp|−d(ΓG,0)−d(ΓG,1)=n(2n+1)−2n−n(n+4)2=3n(n−2)2.

The Hosoya polynomials of the commuting graph on Dn, for odd and even values of n ≥ 3, is obtained in the following result.

Theorem 4

For n ≥ 3, let ΓG be the commuting graph on Γ = Dn, then:

H(ΓG,x)={n2{3(n−1)x2+(n+1)x+4},for odd n,n2{3(n−2)x2+(n+4)x+4},for even n.

Proof

Using the coefficients d (ΓG,k), computed in Propositions 2 and 3, in the formula of the Hosoya polynomial, we have:

For odd n:

H(ΓG,x)=d(ΓG,0)x0+d(ΓG,1)x1+d(ΓG,2)x2=(2n)x0+(n(n+1)2)x+(3n(n−1)2)x2=n2{3(n−1)x2+(n+1)x+4}.

For even n:

H(ΓG,x)=d(ΓG,0)x0+d(ΓG,1)x1+d(ΓG,2)x2=(2n)x0+(n(n+4)2)x1+(3n(n−2)2)x2=n2{3(n−2)x2+(n+4)x+4}.

2.2 Reciprocal status Hosoya polynomial

Firstly, we find the reciprocal status of each vertex of the commuting graph on Dn, for odd and even values of n ≥ 3, in the following two results, respectively.

Proposition 5

If l is a vertex in the commuting graph on Dn for odd values of n ≥ 3, then:

rs(l)={2n−1,whenever l∈ζ(Dn),3n2−1,whenever l∈Ω3,n,whenever l∈Ω2.

Proof

By Proposition 1, the commuting graph on Dn is K1 +(K|Ω3| ∪ N|Ω2|) with the vertex set ζ(Dn)∪Ω3 ∪Ω2. Accordingly, we have:

Whenever l∈ζ(Dn): ecc(l) = 1, and by the definition of reciprocal status, we get

rs(l)=(11)(n+n−1)=2n−1.

Whenever l∈Ω3: ecc(l) = 2, and by the definition of reciprocal status, we get

rs(l)=(11)(n−1)+(12)n=3n2−1.

Whenever l∈Ω2: ecc (l) = 2, and by the definition of reciprocal status, we get

rs(l)=(11)1+(12)(2n−2)=n.

Proposition 6

For even values of n ≥ 4, if l is any vertex of the commuting graph on Dn, then:

rs(l)={2n−1,whenever l∈ζ(Dn),3n2−1,whenever l∈Ω3,n+1,whenever l∈Ω2.

Proof

By Proposition 1, the commuting graph on Dn is K2+(K|Ω3|∪n2K2) with the vertex set ζ(Dn)∪Ω3∪Ω2. Accordingly, we have:

Whenever l∈ζ(Dn): ecc(l) = 1, and by the definition of reciprocal status, we get

rs(l)=(11){1+n−2+(n2)2}=2n−1.

Whenever l∈Ω3: ecc(l) = 2, and by the definition of reciprocal status, we get

rs(l)=(11)(n−3+2)+(12)(n2)2=3n2−1.

Whenever l ∈Ω2: ecc(l) = 2, and by the definition of reciprocal status, we get

rs(l)=(11)3+(12){(n−2)+(2n2−2)}=n+1.

The next two results provide the reciprocal status Hosoya polynomial of the commuting graph on Dn.

Theorem 7

For odd n ≥ 3, if ΓG is the commuting graph on Γ = Dn. Then:

Hrs(ΓG)=(n−1)x7n2−2+nx3n−1+(n−12)x3n−2.

Proof

Proposition 5 implies that there are three types (a ~ b, a ~ c, b ~ b) of edges in ΓG according to the reciprocal statuses of their end vertices, where a = 2n −1, b=3n2−1 , c = n. Table 1 shows the edge partition accordingly.

Table 1

Edge partition of ΓG according to reciprocal statuses

Edges type Partition of edge set Number of edges
a ~ b Ea~b ={uv∈E(ΓG)|rs(u) =a, rs (v)=b} |Ea~b|=n−1
a ~ c Ea~c ={uv ∈E(ΓG)|rs(u)= a, rs(v)=c} |Ea~c|=n
b ~ b Eb~b={uv ∈E(ΓG)|rs(u)= b, rs(v)=b} |Eb~b|=(n−12)

By using the edge partition, given in the Table 1, in the formula of the reciprocal status Hosoya polynomial, we have:

Hrs(G)=∑Ea−bxa+b+∑Ea−cxa+c+∑Eb−bxb+b=(n−1)x2n−1+3n2−1+nx2n−1+n+(n−12)x3n2−1+3n2−1=(n−1)x7n2−2+nx3n−1+(n−12)x3n−2.

Theorem 8

For even n ≥ 4, if ΓG is the commuting graph on Γ = Dn. Then:

Hrs(ΓG)=(n−22)x3n−2+(2n−4)x7n22+x4n−2+2nx3n+n2x2n+2.

Proof

Proposition 6 implies that there are five types (a ~ a, a ~ b, b ~ b, b ~ c, c ~ c) of edges in ΓG accordingly, the edge partition is given in the Table 2, according to the 3n reciprocal statuses of their end vertices, where a=3n2−1 , b = 2n −1, c =n + 1.

Table 2

Edge partition of ΓG according to reciprocal statuses

Type of edges Partition of edge set Number of edges
a ~ a Ea~a={uv∈E(ΓG)|rs(u)=a,rs(v)=a} |Ea~a|=(n−22)
a ~ b Ea~b={uv∈E(ΓG)|rs(u)=a,rs(v)=b} Ea~b=2(n − 2 )
b ~ b Eb~b={uv∈E(ΓG)|rs(u)=b,rs(v)=b} Eb~b=|1|
b ~ c Eb~c={uv∈E(ΓG)|rs(u)=b,rs(v)=c} Eb~c=2n
c ~ c Ec~c={uv∈E (ΓG)|rs(u)=c,rs(v)=c} |Ec−c|=n2

By using the edge partition of ΓG, given in the Table 2, in the formula of the reciprocal Hosoya polynomial, we have:

Hrs(G)=∑Ea~axa+a+∑Ea~bxa+b+∑Eb~bxb+b+∑Eb~cxb+c+∑Ec~cxc+c   =(n−22)x2(3n2−1)+2(n−2)x3n2−1+2n−1+1x2(2n−1)    +2nx2n−1+n+1+(n2)x2(n+1)=(n−22)x3n−2+(2n−4)x7n2−2+x4n−2+2nx3n+(n2)x2n+2.

3 Hosoya index

In this section, we investigate the Hosoya index of the commuting graph of the dihedral group. The largest possible value of the Hosoya index, on a graph with n vertices, is given by the complete graph Kn ( Tichy and Wagner, 2005). Generally, the Hosoya index of a complete graph Kn, n ≥ 1 is:

1+∑k=1[n2]1k∏i=0k−1(n−2i2),

which can be viewed in accordance with the number of non-empty matchings given in Table 3, where mk denotes the number of matchings of cardinality k, 1≤k≤[n2] .

Table 3

The number of non-empty matchings in a complete graph Kn

Kn m1 m2 m3 m4 ⋯ mk
K2 (22)
K3 (32)
K4 (42) 12(42)(22)
K5 (52) 12(52)(32)
K6 (62) 12(62)(42) 13(62)(42)(22)
K7 (72) 12(72)(52) 13(72)(52)(32)
K8 (82) 12(82)(62) 13(82)(62)(42) 14(82)(62)(42)(22)
K9 (92) 12(92)(72) 13(92)(72)(52) 14(92)(72)(52)(32)
⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮
Kn (n2) 12(n2)(n−22) 13(n2)(n−22)(n−42) 14(n2)(n−22)(n−42)(n−62) … 1k∏i=0k−1(n−2i2)

For all odd values of n ≥ 3, the Hosoya index is computed in the following result.

Theorem 9

Let ΓG be a commuting graph on Γ = Dn, for odd n ≥ 3. Then the Hosoya index of ΓG is:

1+(n2)+n+∑k=2[n2][1k∏i=0k−1(n−2i2)+nk−1∏i=0k−2(n−2i−12)].

Proof

As ΓG =K1 +(K|Ω3| ∪ N|Ω2|) with V(ΓG) =ζ(Dn) ∪Ω3 ∪Ω2, by Proposition 1, so there are three types of edges in ΓG:

  • Type-1: u ~ v for u ,v∈Ω3;

  • Type-2: u ~ v for u∈Ω3 and v ∈ζ(Dn);

  • Type-3: u ~ v for u∈Ω2 and v ∈ζ(Dn).

As ζ(Dn)∪Ω3 = Ω1 induces a complete subgraph Kn on n vertices in ΓG, so there are two types of matchings in ΓG:

  • (M1) Matchings of edges of Type-1 and Type-2;

  • (M2) Matchings of edges of Type-1 and Type-3.

The number of matchings of each type can be found as follows:

  • (M1) Since the edges of Type-1 and Type-2 are the edges of complete graph Kn induced by the vertices in ζ(Dn)∪Ω3 = Ω1, so the number of matchings in this type can be obtained by counting the matchings in Kn for all n ≥ 3, which are given in Table 4, where mk denotes the number of matchings of order k for 1≤k≤[n2] .

  • (M2) Each matching of this type can be obtained by adding one edge of Type-3 into each matching of the edges of Type-1. As each edge of Type-1 is an edge of a complete graph Kn−1 induced by the vertices in Ω3, so each matching of the edges of Type-1 is actually the matching in a complete graph Kn−1. The numbers of such matchings are listed in Table 5, where mk denotes the number of matchings of order k for 1≤k≤[n2] .

Table 4

The number of non-empty matchings in a complete graph Kn for odd n ≥ 3

Kn m1 m2 m3 m4 ⋯ mk
K3 (32)
K5 (52) 12(52)(32)
K7 (72) 12(72)(52) 13(72)(52)(32)
K9 (92) 12(92)(72) 13(92)(72)(52) 14(92)(72)(52)(32)
⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮
Kn (n2) 12(n2)(n−22) 13(n2)(n−22)(n−42) 14(n2)(n−22)(n−42)(n−62) ⋯ 1k∏i=0k−1(n−2i2)
Table 5

The number of non-empty matchings in a complete graph Kn−1 for odd n ≥ 3

Kn m1 m2 m3 m4 ⋯ mk
K2 (22)
K4 (42) 12(42)(22)
K6 (62) 12(62)(42) 13(62)(42)(22)
K8 (82) 12(82)(62) 13(82)(62)(42) 14(82)(62)(42)(22)
⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮
Kn−1 (n−12) 12(n−12)(n−32) 13(n−12)(n−32)(n−52) 14(n−12)(n−32)(n−52)(n−72) ⋯ 1k−1∏i=0k−2(n−2i−12)

Now, since there are n edges of Type-3, so the required matchings can be obtained as follows:

  • Matchings of order 1: These are n such matchings corresponding to n edges of Type-3.

  • Matchings of order 2: Each of these matchings can be obtained by adding one edge of Type-3 into each matching of order 1 in Kn−1. There are n edges of Type-3 and (n−12) matchings of order 1 in Kn−1, by Table 5. Therefore, by the rule of product, the number of matchings of order 2 is:

    n(n−12).

  • Matchings of order 3: Each of these matchings can be obtained by adding one edge of Type-3 into each matching of order 2 in Kn−1. There are n edges of Type-3 and 12(n−12)(n−32) matchings of order 2 in Kn−1, by Table 5. Therefore, by the rule of product, the number of matchings of order 3 is:

    12n(n−12)(n−32).

  • Matchings of order 4: Each of these matchings can be obtained by adding one edge of Type-3 into each matching of order 3 in Kn−1. There are n edges of Type-3 and

    13(n−12)(n−32)(n−52)

    matchings of order 3 in Kn−1, by Table 5. Therefore, by the rule of product, the number of matchings of order 4 is:

    13n(n−12)(n−32)(n−52).

  • Matchings of order k : Generally, each matching of order k can be obtained by adding one edge of Type-3 into each matching of order k − 1 in Kn−1. There are n edges of Type-3 and

    1k−1∏i=0k−2(n−2i−12).

    matchings of order k −1 in Kn−1, by Table 5. Therefore, by the rule of product, the number of matchings of order k is:

    nk−1∏i=0k−2(n−2i−12).

Now, by the rule of sum, the total number of matchings (matchings (M1) + matchings (M2)) in ΓG of each order can be counted as follows:

  • The number of matching of order 1 is:

    (n2)+n.

  • The number of matching of order 2 is:

    12(n2)(n−22)+n(n−12).

  • The number of matching of order 3 is:

    13(n2)(n−22)(n−42)+12n(n−12)(n−32).

  • The number of matching of order 4 is:

    14(n2)(n−22)(n−42)(n−62)+13n(n−12)(n−32)(n−52).

  • Generally, the number of matching of order k is:

    1k∏i=0k−1(n−2i2)+nk−1∏i=0k−2(n−2i−12),

    where 2≤k≤[n2] .

  • Hence, the Hosoya index of ΓG is:

    1+(n2)+n+∑k=2[n2][1k∏i=0k−1(n−2i2)+nk−1∏i=0k−2(n−2i−12)].

When n = 2, then Γ = D2 is an abelian group and so the commuting graph ΓG is the complete graph K4. Hence, by Table 3, the Hosoya index of ΓG is 1 + m1 +m2 = 10. The following result provides the Hosoya index of the commuting graph of Dn for even values of n > 3.

Theorem 10

Let ΓG be a commuting graph on Γ = Dn for even n > 3. Then the Hosoya index of ΓG is:

1+∑k=1n2mk1+∑k=12mk2+∑k=1n2mk3+∑k=2n2+1mk4+m25+∑k=2n2mk6+∑k=2nmk7,

where:

mk1=1k∏i=0k−1(n−2i2),

m12=2n,m22=n(n−1),

mk3=(n2k),

m24=2n(n−22),

mk4=n[2k−1∏i=0k−2(n−2i−22)+n−1k−2∏i=0k−3(n−2i−22)]for 3≤k≤n2,

mn2+14=n(n−1)k−2∏i=0k−3(n−2i−22),

m25=2n(n−1),

m26=2n(n2−1),

mk6=n[2(n2−1k−1)+(n2−1k−2)+2(n−2)(n2−2k−2)]for 3≤k≤n2,

mk7=∑j=1k−11j∏i=0j−1(n−2i2)(n2k−j)for2≤k≤n.

Proof

By Proposition 1, ΓG=K2+(K|Ω3|∪n2K2) with V(ΓG) =ζ(Dn)∪Ω3∪Ω2, where Ω2=∪i=0n2−1Ω2i . Thus, we have following types of edges in ΓG.

  • Type-1: x ~ y for x,y ∈ Ω3;

  • Type-2: x ~ y for x,y ∈ ζ(Dn);

  • Type-3: x ~ y for x ∈ Ω3 and y ∈ζ(Dn);

  • Type-4: x ~ y for x ∈ Ω2 and y ∈ ζ (Dn);

  • Type-5: x ~ y for x, y∈Ω2i⊆Ω2 , where 0≤i≤n2−1 .

According to these types, following seven types of matchings between the edges of ΓG exist:

  • (m1) Matchings between the edges of Type-1, Type-2 and Type-3;

  • (m2) Matchings between the edges of Type-4;

  • (m3) Matchings between the edges of Type-5;

  • (m4) Matchings between the edges of Type-1 and Type-4;

  • (m5) Matchings between the edges of Type-3 and Type-4;

  • (m6) Matchings between the edges of Type-4 and Type-5;

  • (m7) Matchings between the edges of Type-1, Type-2 Type-3 and the edges of Type-5.

The number of all these types of matchings is computed as follows:

  • (m1) Since ζ(Dn)∪Ω3=Ω1 induces a complete graph kn, so the edges of Type-1, Type-2 and Type-3 are the edges of kn, and all the matchings between these edges are counted in Table 6, where mk1 denotes the number of matchings of order k, where 1≤k≤n2 .

  • (m2) Let mk2 denotes the number of matchings of order k, for k = 1,2.

  • For m12 : The number of matchings of order 1 is equal to the number of edges of Type-4, which is 2n. Hence m12=2n .

  • For m22 : Let e =x ~ y be an edge of Type-4 with x∈Ω2i for fixed 0≤i≤n2−1 and y ∈ζ(Dn). Then, together with the edge e, each edge of Type-4 having one end in Ω2 − {x} and one end in ζ(Dn) − {y} form a matching of order 2. Therefore m22=12(4(n−1)×n2)=n(n−1) . There is no matching of order greater than two in this case.

  • (m3) There are n2 edges of Type-5, and no two of them 2 shares a common vertex. Therefore, there exists a matching of every order k such that 1≤k≤[n2] . Let mk3 denotes the number of matchings of order k. Then:

    mk3=(n2k).

  • (m4) Let mk4 denotes the number of matchings of order k, for 1≤k≤n2+1 . Then m14=0 in this case. In Γ, no edge of Type-1 shares a common vertex with any edge of Type-4. Therefore, in this case we can get a matching by combining any matching of the edges of Type-1 with each matching of the edges of Type-4. As there are mji matchings of order j, 1≤j≤n−22 , between the edges of Type-1, which are the edges of a complete graph Kn−2, where each mj1 can be obtained from Table 6 and there are m12=2n and m22=n(n−1) matching of order 1 and 2, respectively, between the edges of Type-4. Therefore, by the rule of product, we get:

    m24=m12×m11=2nm11,

    for 3≤k≤n2 :

    mk4=m12×mk−11×m22×mk−21=2nmk−11+n(n−1)mk−21=n(2mk−11+(n−1)mk−21),

    and for k=n2+1 :

    mk4=m22×mk−21=n(n−1)mn2−11.

  • (m 5) Let mk5 denotes the number of matchings of order k for k = 1,2. Then m15=0 in this case. Here, we can use matchings of order 1 only between the edges of Type-4. For otherwise, we cannot use any edge of Type-3 because both types of edges commonly shared the vertices in ζ(Dn). Hence, we can get matchings of order 2 only in this case. Let M ={e =x ~ y} be a matching of order 1 between the edges of Type-4 with x∈Ω2i for 0≤i≤n2−1 and y ∈ζ(Dn). Then every edge of Type-3, not adjacent with y can contribute to form a matching of order 2. Since there are n − 2 such edges of Type-3 each of which can be used in any of 2n matchings of order 1 between the edges of Type-4, so by the rule of product, we have:

    m25=2n(n−2).

  • (m6) Let mk6 denotes the number of matchings of order k for 1≤k≤n2 . Then m16=0 in this case, because to find matchings, both the matchings of order 1 and 2 between the dges of Type-4 will be used, and any matching of order j, 1≤j≤n2−1 between the edges of Type-5 will be used. Accordingly, by counting these matchings with the rule of product, we get:

    m26=4×1×(n2−11)×n2=2n(n2−1),

    and for 3≤k≤n2 ,

    mk6=n[2(n2−1k−1)+(n2−1k−2)+2(n−2)(n2−2k−2)].

  • (m7) Since the edges of Type-1, Type-2 and Type-3 are the edges of a complete graph kn induced by ζ(Dn)∪Ω3 = Ω1, so in this case we find the matchings between the edges of Type-5 and the edges of kn. Let mk7 be the number of matchings of order k. Then m17=0 . Since no edge of Type-5 share a common vertex with any edge of kn, so corresponds to each matching of the edges of Type-5, every matching of the edges of kn can be used to find a matching in this case. As there are m1t matchings of order 1≤t≤n2 between the edges of kn, counted in Table 6, and there are mj3=(n2j) matchings of order 1≤j≤n2 between the edges of Type-5. Thus, 2 the maximum order of a matching in this case is n2+n2=n . Therefore, we can find mk7 for 2 ≤ k ≤ n as follows:

    m27=m11m13,m37=m11m23+m21m13,m47=m11m33+m21m23+m31m13,m57=m11m43+m21m33+m31m23+m41m13,

    and so on, generally, we have:

    mk7=∑j=1k−1mj1mk−j3.

  • Hence, by the rule of sum, the Hosoya index of ΓG is: 1+∑i=17 the number of matchings (mi)

    =1+∑k=1n2mk1+∑k=12mk2+∑k=1n2mk3+∑k=2n2+1mk4+m25+∑k=2n2mk6+∑k=2nmk7,

    with

    mk1=1k∏i=0k−1(n−2i2),m12=2n,m22=n(n−1),mk3=(n2k),m24=2n(n−22),mk4=n[2k−1∏i=0k−2(n−2i−22)+n−1k−2∏i=0k−3(n−2i−22)]for 3≤k≤n2,mn2+14=n(n−1)k−2∏i=0k−3(n−2i−22),m25=2n(n−1),

    m26=2n(n2−1),mk6=n[2(n2−1k−1)+(n2−1k−2)+2(n−2)(n2−2k−2)]for 3≤k≤n2,mk7=∑j=1k−1∏i=0j−1(n−2i2)(n2k−j)for 2≤k≤n.

Table 6

The number of non-empty matchings in a complete graph Kn for even n ≥ 2

Kn m11 m21 m31 m41 … mk1
K2 (22)
K4 (42) 12(42)(22)
K6 (62) 12(62)(42) 13(62)(42)(22)
K8 (82) 12(82)(62) 13(82)(62)(42) 14(82)(62)(42)(22)
⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮
Kn−1 (n2) 12(n2)(n−22) 13(n2)(n−22)(n−42) 14(n2)(n−22)(n−42)(n−62) … 1k∏i=0k−1(n−2i2)

  1. Funding information:

    This work is truly self-supported. Authors state no funding involved.

  2. Author contributions:

    Ghulam Abbas: writing, analysis, methodology; Anam Rani: formal analysis, writing – review and editing; Muhammad Salman: supervision, project administration, data analyses; Tahira Noreen: writing – initial draft, computation; Usman Ali: conceptualization, visualization, investigation.

  3. Conflict of interest:

    Authors state no conflict of interest.

  4. Data availability statement:

    All kinds of data and materials, used to compute the results, are provided in sections 1. We have no further source of data and materials to share.

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Received: 2021-01-18
Accepted: 2021-03-13
Published Online: 2021-06-18

© 2021 Ghulam Abbas et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/mgmc-2021-0017
Keywords for this article
geodesic; Hosoya polynomial; reciprocal Hosoya polynomial; Hosoya index; molecular struture; network
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