(β ,α)Connectivity Index of Graphs

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Abstract

Let Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined as

χαβ(G)=Σv1,v2vβ+1Eβ(G)(dG(v1)dG(v2)...dG(vβ+1))α.

The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.

Abstract

Let Eβ (G) be the set of paths of length β in a graph G. For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index is defined as

χαβ(G)=Σv1,v2vβ+1Eβ(G)(dG(v1)dG(v2)...dG(vβ+1))α.

The (2,1)-connectivity index shows good correlation with acentric factor of an octane isomers. In this paper, we compute the (2, α)-connectivity index of certain class of graphs, present the upper and lower bounds for (2, α)-connectivity index in terms of number of vertices, number of edges and minimum vertex degree and determine the extremal graphs which achieve the bounds. Further, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.

1 Introduction

Let G = (V,E) be a simple graph with n = |V| vertices and m = |E|edges. As usual, nis said to be an order and mthe size of G. The subdivision graph S(G) is the graph obtained from Gby replacing each edge by a path of length 2. The line graph L(G) of Gis the graph whose vertex set is E(G) in which two vertices are adjacent if and only if they are adjacent in G. The tadpole graph Tn,k is the graph obtained by joining a cycle of nvertices with a path of length k. The cartesian product G × Hof graphs Gand Hhas the vertex set V(G × H) = V(G) × V(H) and (a,x)(b,y) is an edge of G × Hif and only if [a = band xy ∈ E(H)] or [x = yand ab ∈ E(G)]. The ladder graph Ln is given by Ln = K2 × Pn, where Pn is the path of length n. Let Eβ (G) be the set of paths of length βin G. We refer to [13] for unexplained terminology and notation.

Chemical graph theory is a branch of mathematical chemistry concerned with the study of chemical graphs. Chemical graphs are models of molecules in which atoms are represented by vertices and chemical bonds by edges of a graph. A graphical invariant is a number related to a graph. In other words, it is a fixed number under graph automorphisms. In chemical graph theory, these invariants are also called the topological indices. The first and second Zagreb indices are defined as

M1(G)=ΣuV(G)dG(u)2=ΣuvE(G)[dG(u)+dG(v)]andM2(G)=ΣuV(G)dG(u)dG(v)

respectively.

The connectivity index of an organic molecule whose molecular graph Gis defined (see [12, 17]) as

1χα(G)=uvE(G)(dG(u)dG(v))α.

For an integer β ≥ 1 and a real number α, the (β,α)-connectivity index [14] is defined as

χαβ(G)=Σv1,v2vβ+1Eβ(G)(dG(v1)dG(v2)...dG(vβ+1))α.

The higher connectivity indices are of great interest in molecular graph theory [15, 22] and some of their mathematical properties have been reported in [1, 18, 19]. Chemical applications of higher connectivity indices are the motivations for our study.

By Eq. (1), it is consistent to define (2, α)-connectivity index and (2,1)-connectivity index as

χα2(G)=ΣuvwE2(G)(dG(u)dG(v)dG(w))αand

χ12(G)=ΣuvwE2(G)(dG(u)dG(v)dG(w))

respectively.

The present paper is organized as follows. In Section 2, we study the chemical applicability of the (2,1)-connectivity index. In Section 3, we compute the (2,α)-connectivity index of certain class of graphs and present the upper and lower bounds for (2, α)-connectivity index in terms of the number of vertices, the number of edges and the minimum vertex degree and determine the extremal graphs which achieve the bounds. In Section 4, we compute the (2, α)-connectivity index of line graphs of subdivision graphs of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q], tadpole graphs, wheel graphs and ladder graphs.

2 On the chemical applicability of the (2,1)-connectivity index

Octane isomers have become an important set of organic molecules to test the applicability of various topological parameters in quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR). The productivity of (2,1)-connectivity index was tested using a dataset of octane isomers, found at https://www.moleculardescriptors.eu/dataset.htm. It is shown that the (2,1)-connectivity index has a good correlation with the acentric factor of an octane isomers.

The dataset of octane isomers (first two columns of Table 1) are taken from https://www.moleculardescriptors.eu/dataset.htm and the last column of Table 1 is computed from the definition of (2,1)-connectivity index.

Table 1

Experimental values of the acentric factor and corresponding values of 2χ1 of octane isomers.

AlkaneAcentFac2χ1
n-octane0.39789840
2-methyl-heptane0.37791647
3-methyl-heptane0.37100254
4-methyl-heptane0.37150456
3-ethyl-hexane0.36247264
2,2-dimethyl-hexane0.33942664
2,3-dimethyl-hexane0.34824770
2,4-dimethyl-hexane0.34422363
2,5-dimethyl-hexane0.3568354
3,3-dimethyl-hexane0.32259680
3,4-dimethyl-hexane0.34034578
2-methyl-3-ethyl-pentane0.33243381
3-methyl-3-ethyl-pentane0.30689996
2,2,3-trimethyl-pentane0.30081696
2,2,4-trimethyl-pentane0.3053775
2,3,3-trimethyl-pentane0.293177103
2,3,4-trimethyl-pentane0.31742287
2,2,3,3-tetramethylbutane0.255294120

The linear regression model for the acentric factor of Table 1 is obtained by using the least squares fitting procedure as implemented in R−software [24]. The fitted model is

AcentFac=0.45473760.00161252χ1

The values of (2,1)-connectivity index against values of acentric factor of an octane isomers are plotted in Fig. 1. The absolute value of correlation coefficient between AcentFac and 2χ1 is 0.95802 and with standard error 0.01047.

Fig. 1
Fig. 1

Scatter diagram of AcentFacon 2χ1, superimposed by the fitted regression line.

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00003

3 Estimating the (2,α)-connectivity index of graphs

We start by stating the following observation, which is needed to prove our main results.

Remark 1. For a graph Gon medges, the number of paths of length 2 in Gis m+12M1(G).

Theorem 1

For a path Pn on n > 4 vertices,

2χα(Pn)=24α+8α(n4)

Proof.For a path Pn on n >4 vertices each vertex is of degree either 1 or 2. Based on the degree of vertices on the path of length 2 in Pn we can partition E2(Pn). In Pn, path (1,2,2) appears 2 times and path (2,2,2) appears (n −4) times. Hence by Eq. (2) we get the required result.

Theorem 2

For a wheel graph Wn+1,

χα(Wn+1)=27α2n+(9n)αn2+3n2.

Proof.For a wheel Wn+1 on n ≥3 vertices each vertex is of degree either 3 or n. Based on the degree of vertices on the path of length 2 in Wn+1 we can partition E2(Wn+1). In Wn+1, path (3,3,3) appears ntimes and path (3,3,n) appears n2+3n n2+3n2 times. Therefore by Eq. (2), we get the required result.

Theorem 3

For a complete bipartite graph Kr,s,

χα2(Kr,s)=r2α+1sα+1(s1)2+rα+1s2α+1(r1)2

Proof.For a complete bipartite graph Kr,s on r + svertices each vertex is of degree either ror s. Based on the degree of vertices on the path of length 2 in Kr,s we can partition E2(Kr,s). In Kr,s, path (r,s,r) appears rs(s1)2 times and path (s,r,s) appears sr(r1)2 times. Hence by Eq. (2), we get the required result.

Theorem 4

Let G be a r-regular graph on n vertices,

χα2(G)=r3α+1n(r1)2.

Proof.Since Gis a r-regular graph, the path (r,r,r) appears nr(r1)2

times in G. Therefore by Eq. (2), we get the required result.

Corollary 5

For a cycle Cn, 2χa(Cn) = 8a · n.

Corollary 6

For a complete graph Kn, χα2(Kn)=n(n2)(n1)3α+12.

Lemma 7

(c.f. [3]). Let G be a graph with n vertices and m edges. Then

M1(G)m(2mn1+n2).

Lemma 8

(c.f. [4]). Let G be a graph with n vertices and m edges, m > 0. Then the equality

M1(G)=m(2mn1+n2)

holds if and only if G is isomorphic to star graph Sn or Kn or Kn−1K1.

Theorem 9

Let G be a graph with n vertices and m edges. Then

χ12(G)(n1)3αm(mn1+n42)

the equality holds if and only if G is isomorphic to Kn.

Proof.

χα2(G)=ΣuvwE2(G)[dG(u)dG(v)dG(w)]αΣuvwE2(G)(n1)3α

=(n1)3α(m+12M1(G))(n1)3α(m+12m(2mn1+n2))=(n1)3αm(mn1+n42).

Relations (6) and (7) were obtained by taking into account for each vertices v ∈ V(G), we have dG(v) n − 1 and Eq. (4), respectively.

Suppose that equality in (5) holds. Then inequalities (6) and (7) become equalities. From (6) we conclude that for every vertex v, dG (v) = n −1. Then from Eq. (7) and Lemma 8 it follows that G is a complete graph. Conversely, let G be a complete graph. Then it is easily verified that equality holds in (5).

Lemma 10

(c.f. [4]). Let G be a graph with n vertices and m edges. Then

M1(G)2m(2p+1)pn(1+p),wherep=2mn,

and the equality holds if and only if the difference of the degrees of any two vertices of graph G is at most one.

Theorem 11

Let G be a graph with n vertices, m edges and the minimum vertex degree δ. Then

2χα(G)δ3α2(4mppn(p+1)),wherep=2mn,

and the equality holds if and only if G is a regular graph.

Proof.

χα2(G)=ΣuvwE2(G)[dG(u)dG(v)dG(w)]αΣuvwE2(G)δ3α

=δ3α(m+12M1(G))δ3α(m+12(2m(2p+1)pn(1+p)))=δ3α2(4mppn(p+1)).

Relations (9) and (10) were obtained by taking into accounting for each vertices v ∈ V(G), we have dG(v) ≥ δ and Eq. (8), respectively.

Suppose now that equality in (8) holds. Then inequalities (9) and (10) become equalities. From (9) we conclude that for every vertex v, dG(v) = δ. Then from Eq. (10) and Lemma 10 it follows that Gis a regular graph. Conversely, let Gbe a regular graph. Then it is easily verified that equality holds in (8).

4 Computing the (2,α)-connectivity index of line graphs of subdivision graphs of some families of graphs

In [16], Nadeem et al. obtained expressions for certain topological indices of the line graphs of subdivision graphs of 2D-lattice, nanotube, and nanotorus of TUC4C8[p,q], where pand qdenote the number of squares in a row and the number of rows of squares, respectively in 2D-lattice, nanotube and nanotorus as shown in Figure 2 (a), (b) and (c)respectively. The numbers of vertices and edges of 2D-lattice, nanotube and nanotorus of TUC4C8[p,q] are given in Table 2. Readers interested in other information on nanostructures can be referred to [2, 59, 11].

Fig. 2
Fig. 2

(a) 2D-lattice of TUC4C8[4,3]; (b) TUC4C8[4,3] nanotube; (c) TUC4C8[4,3] nanotorus.

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00003

Table 2

Number of vertices and edges.

GraphNumber of verticesNumber of edges
2D-lattice of TUC4C8 [p,q]4pq6pqpq
TUC4C8[p,q] nanotube4pq6 pqp
TUC4C8[p,q] nanotorus4pq6pq

In [20, 21], Ranjini et al. presented explicit formula for computing the Shultz index and Zagreb indices of the subdivision graphs of the tadpole, wheel and ladder graphs. In 2015, Su and Xu [23] calculated the general sum-connectivity index and coindex of the L(S(Tn,k)), L(S(Wn)) and L(S(Ln)). In [11], Nadeem et al. derived some exact formulas for ABC4 and GA5 indices of the line graphs of the tadpole, wheel and ladder graphs by using the notion of subdivision.

Lemma 12

(c.f. [16]). Let X be the line graph of the subdivision graph of 2D-lattice of TUC4C8[p,q]. Then M1(X) = 108pq −38p −38q.

Theorem 13

Let X be the line graph of the subdivision graph of 2D-lattice of TUC4C8[p,q]. Then 2χα(X) = 8α+1 + 4(12α + 2 ·18α)(p + q −2) + 27α(36pq −26p −26q + 16).

Proof.The subdivision graph of 2D-lattice of TUC4C8[p,q] and the graph Xare shown in Fig. 3 (a) and (b), respectively. In Xthere are total 2(6pq− p−q) vertices each vertex is of degree either 2 or 3 and 18pq−5p−5qedges. From Observation 1 and Lemma 12, we get 36pq −14p −14qof paths of length 2 in X. Based on the degree of vertices on the path of length 2 in Xwe can partition E2(X) as shown in Table 3. Apply Eq. (2) to Table 3 and get the required result.

Fig. 3
Fig. 3

(a) Subdivision graph of 2D-lattice of TUC4C8[4,3]; (b) Line graph of the subdivision graph of 2D-lattice of TUC4C8[4,3].

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00003

Table 3

Partition of paths of length 2 of the graph X.

(dx(u), dX (v), dX (w)) where uvwE2(X)(2,2,2)(2,2,3)(3,3,2)(3,3,3)
Number of paths of length 2 in X84(p + q — 2)8(p + q — 2)(36 pq — 26 p — 26q + 16)

Lemma 14

(c.f. [16]). Let Y be the line graph of the subdivision graph of TUC4C8[p,q] nanotube. Then M1(Y) = 108pq −38p.

Theorem 15

Let Y be the line graph of the subdivision graph of TUC4C8[p,q] nanotube. Then 2χα(Y) = 12α · 4p + 18α · 8p + 27α(36pq − 26p).

Proof.The subdivision graph of TUC4C8[p,q] nanotube and the graph Y are shown in Fig. 4 (a) and (b), respectively. In Y there are total 12pq − 2p vertices in which each vertex is of degree either 2 or 3 and 18pq − 5p edges. From Remark 1 and Lemma 14, we get 36pq − 14p number of paths of length 2 in Y. Based on the degree of vertices on the paths of length 2 in Y we can partition E2(Y) as shown in Table 4. Apply Eq. (2) to Table 4 and get the required result.

Fig. 4
Fig. 4

(a) Subdivision graph of TUC4C8[4,3] of nanotube; (b) line graph of the subdivision graph of TUC4C8[4,3] of nanotube.

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00003

Table 4

Partition of paths of length 2 of the graph Y.

(dY (u), dY(v), dY (w)) where uvwE2(Y)(2,2,3)(3,3,2)(3,3,3)
Number of paths of length 2 in Y4p8p(36pq — 26p)

Theorem 16

Let Z be the line graph of the subdivision graph of TUC4C8[p,q] nanotorus. Then 2χα(Z) = 27α · 36pq.

Proof.The subdivision graph of TUC4C8[p,q] nanotorus and the graph Zare shown in Fig. 5 (a) and (b), respectively. Since Zis a 3-regular graph with 12pq vertices and 18pq edges. Therefore by Theorem 4, we get the required result.

Fig. 5
Fig. 5

(a) Subdivision graph of TUC4C8[4,3] of nanotorus; (b) Line graph of the subdivision graph of TUC4C8[4,3] of nanotorus.

Citation: Applied Mathematics and Nonlinear Sciences 2, 1; 10.21042/AMNS.2017.1.00003

Lemma 17

(c.f. [20, 23]). Let A be a line graph of the subdivision graph of the tadpole graph Tn,k. Then M1(A) = 8n + 8k + 12.

Theorem 18

Let A be a line graph of the subdivision graph of the tadpole graph Tn,k. Then

2χαA=4α+18α6+12α3+27α3+8α2n+2k8fork>1.9α+18α4+12α2+27α3+8α2n4fork=1.

Proof.First of all, we consider graph A for n ≥ 3 and k > 1. In this graph there are total 2(n + k) vertices and 2n + 2k + 1 edges. From Remark 1 and Lemma 17, we get 2k + 2n + 5 of paths of length 2 in A. Based on the degree of vertices on the paths of length 2 in A we can partition E2(A) as shown in Table 6. Apply Eq. (2) to Table 6 and get the required result. By similar arguments we can obtain the expression of 2χα(A) for k = 1 from Table 5.

Table 5

Partition of paths of length 2 of the graph A = L(S(Tn,k)) for k = 1

(dA(u), dA(v), dA (w)) where uvwE2(A)(1,3,3)(2,3,3)(2,2,3)(3,3,3)(2,2,2)
Number of paths of length 2 in A24232n − 4

Table 6

Partition of paths of length 2 of the graph A = L(S(Tn,k)) for k > 1

(dA(u), dA(v), dA(w)) where uvw ∈ E2(A)(1,2,2)(2,3,3)(2,2,3)(3,3,3)(2,2,2)
Number of paths of length 2 in A16332n + 2k – 8

Lemma 19

(c.f. [20]). Let B be a line graph of the subdivision graph of the wheel graph Wn+1. Then M1(B) = n3 + 27n.

Theorem 20

Let B be a line graph of the subdivision graph of the wheel graph Wn+1. Then χα2(B)=7n27α+nα+19α2+3αn2α+1(n1)+n3α+1n(n1)(n2)2.

Proof.The graph L(S(Wn+1)) contains 4(n + 1) vertices and n2+9n2 edges. From Remark 1 and Lemma 19, we get n3n2+18n2 number of paths of length 2 in B. Based on the degree of vertices on the paths of length 2 in Bwe can partition E2(B) as shown in Table 7. Apply Eq. (2) to Table 7 and get the required result.

Table 7

Partition of paths of length 2 of the graph B.

(dB(u),dB(v),dB(w)) where uvw e E2(B)(3,3,3)(3,3,n)(3,n,n)(n,n,n)
Number of paths of length 2 in B7n2nn(n —1)n(n1)(n2)2

Lemma 21

(c.f. [20, 23]). Let C be a line graph of subdivision graph of a ladder graph with order n. Then M1(Z) = 54n −76.

Theorem 22

Let C be a line graph of subdivision graph of a ladder graph with order n. Then 2χα(C) = 8α ·4 + 12α ·4 + 18α ·8 + 27α(18n −44).

Proof.The graph L(S(Ln)) contains 6n −4 vertices and 18n202 edges. From Remark 1 and Lemma 21, we get 18n −28 number of paths of length 2 in C. Based on the degree of vertices on the paths of length 2 in Cwe can partition E2(C) as shown in Table 8. Apply Eq. (2) to Table 8 and get the required result.

Table 8

Partition of paths of length 2 of the graph C.

(dC(u),dC(v),dC(w))where uvwe E2(C)(2,2,2)(2,2,3)(2,3,3)(3,3,3)
Number of paths of length 2 in C44818n — 44

Communicated by Wei Gao
Acknowledgements

This research is supported by UGC-SAP DRS-III, New Delhi, India for 2016-2021: F.510/3/DRS-III/2016(SAPI) Dated: 29thFeb. 2016.

This research is supported by UGC- National Fellowship (NF) New Delhi. No. F./2014-15/NFO-2014-15-OBCKAR-25873/(SA-III/Website) Dated: March-2015.

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If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    O. Araujo and J.A. de la Peña (1998) The connectivity index of a weighted graph Linear Algebra and its Applications 283 No 1-3 171-177.

    • Crossref
    • Export Citation
  • [2]

    A. R. Ashrafi and S. Yousefi (2007) Computing the Wiener Index of a TUC4C8\(S) Nanotorus MATCH Communications in Mathematical and in Computer Chemistry 57 No 2 403-410.

  • [3]

    D. de Caen (1998) An upper bound on the sum of squares of degrees in a graph Discrete Mathematics 185 No 1-3 245-248.

    • Crossref
    • Export Citation
  • [4]

    K. Ch. Das (2003) Sharp bounds for the sum of the squares of the degrees of a graph Kragujevac Journal of Mathematics 25 31-49.

  • [5]

    M. V. Diudea and E. C. Kirby (2001) The energetic stability of tori and single- wall tubes Fullerene Science and Technology 9 No 4 445-465.

    • Crossref
    • Export Citation
  • [6]

    M. V. Diudea (2002) Hosoya Polynomial in Tori MATCH Communications in Mathematical and in Computer Chemistry 45 109-122.

  • [7]

    M. V. Diudea (2002) Toroidal Graphenes from4−Valent Tori Bulletin of the Chemical Society of Japan 75 No 3 487-492.

    • Crossref
    • Export Citation
  • [8]

    M. V. Diudea B. Parv and E. C. Kirby (2003) Azulenic Tori MATCH Communications in Mathematical and in Computer Chemistry 47 53-70.

  • [9]

    M. V. Diudea M. Stefu Basil Pârv and P. E. John (2004) Wiener Index of Armchair Polyhex Nanotubes Croatica Chemica Acta 77 No 1-2 111-115.

  • [10]

    T. Došlić B. Furtula A. Graovac I. Gutman S. Moradi and Z. Yarahmadi (2011) On Vertex-Degree-Based Molecular Structure Descriptors MATCH Communications in Mathematical and in Computer Chemistry 66 No 2 613-626.

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Figures
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    Scatter diagram of AcentFacon 2χ1, superimposed by the fitted regression line.

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    (a) 2D-lattice of TUC4C8[4,3]; (b) TUC4C8[4,3] nanotube; (c) TUC4C8[4,3] nanotorus.

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    (a) Subdivision graph of 2D-lattice of TUC4C8[4,3]; (b) Line graph of the subdivision graph of 2D-lattice of TUC4C8[4,3].

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    (a) Subdivision graph of TUC4C8[4,3] of nanotube; (b) line graph of the subdivision graph of TUC4C8[4,3] of nanotube.

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    (a) Subdivision graph of TUC4C8[4,3] of nanotorus; (b) Line graph of the subdivision graph of TUC4C8[4,3] of nanotorus.

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