Computation of Topological Indices of Dutch Windmill Graph

Affiliation(s)

^{1}
Post Graduate Department of Mathematics, Maharani’s Science College for Women, Mysore, India.

^{2}
Department of Mathematics, The National Institute of Engineering, Mysore, India.

^{3}
Research and Development Centre, Bharathiar University, Coimbatore, India.

ABSTRACT

In this paper, we compute Atom-bond connectivity index, Fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, Geometric-arithmetic connectivity index and Fifth geometric-arithmetic connectivity index of Dutch windmill graph.

In this paper, we compute Atom-bond connectivity index, Fourth atom-bond connectivity index, Sum connectivity index, Randic connectivity index, Geometric-arithmetic connectivity index and Fifth geometric-arithmetic connectivity index of Dutch windmill graph.

KEYWORDS

ABC Index, ABC_{4} Index,
Sum Connectivity Index,
Randic Connectivity Index,
GA Index,
GA_{5} Index

ABC Index, ABC

Received 11 January 2016; accepted 5 April 2016; published 8 April 2016

1. Introduction

The Dutch windmill graph is denoted by and it is the graph obtained by taking m copies of the cycle with a vertex in common. The Dutch windmill graph is also called as friendship graph if. i.e., friendship graph is the graph obtained by taking m copies of the cycle with a vertex in common. Dutch windmill graph contains vertices and mn edges as shown in the Figures 1-3.

All graphs considered in this paper are finite, connected, loop less and without multiple edges. Let be a graph with n vertices and m edges. The degree of a vertex is denoted by and is the number of vertices that are adjacent to u. The edge connecting the vertices u and v is denoted by uv. Using these terminologies, certain topological indices are defined in the following manner.

Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariants.

The atom-bond connectivity index, ABC index was one of the degree-based molecular descripters, which was introduced by Estrada et al. [1] in late 1990’s. Some upper bounds for the atom-bond connectivity index of

Figure 1..

Figure 2..

Figure 3..

Definition 1.1. Let be a molecular graph and is the degree of the vertex u, then ABC index

of G is defined as,.

The fourth atom bond connectivity index, index was introduced by M. Ghorbani et al. [9] in 2010. Further studies on index can be found in [10] [11] .

Definition 1.2. Let G be a graph, then its fourth ABC index is defined as, ,

where is sum of the degrees of all neighbours of vertex u in G. In other words, , Similarly

for.

The first and oldest degree based topological index was Randic index [12] denoted by and was introduced by Milan Randic in 1975.

Definition 1.3. For the graph G Randic index is defined as,.

Sum connectivity index belongs to a family of Randic like indices. It was introduced by Zhou and Trinajstic [13] . Further studies on Sum connectivity index can be found in [14] [15] .

Definition 1.4. For a simple connected graph G, its sum connectivity index is defined as,

.

The Geometric-arithmetic index, index of a graph G was introduced by D. Vukicevic et al. [16] . Further studies on GA index can be found in [17] - [19] .

Definition 1.5. Let G be a graph and be an edge of G then,.

The fifth Geometric-arithmetic index, was introduced by A.Graovac et al. [20] in 2011.

Definition 1.6. For a Graph G, the fifth Geometric-arithmetic index is defined as,

Where is the sum of the degrees of all neighbors of the vertex u in G, similarly for.

2. Main Results

Theorem 2.1. The Atom bond connectivity index of Dutch windmill graph is.

Proof. Consider the Dutch windmill graph. We partition the edges of into edges of the type where uv is an edge. In we get edges of the type and. Edges of the type and are colored in red and black respectively as shown in the figure [18] . The number of edges of these types are given in the Table 1.

We know that

i.e.,

□Theorem 2.2. The Randic Index of Dutch windmill graph is Proof. We know that

Table 1. Edge partition based on degrees of end vertices of each edge.

Figure 4..

i.e.,

. □

Theorem 2.3. The Geometric-arithmetic index (GA) of Dutch windmill graph is

.

Proof. We know that

. □

Theorem 2.4. The Sum connectivity index of Dutch windmill graph is.

Proof. We know that

i.e.,

. □

Theorem 2.5. The fourth atom bond connectivity index of Dutch windmill graph is

Proof. Any Dutch windmill graph contains vertices and mn edges. Let denote the degree of the vertex u. We partition the edges of into edges of the type where uv is an edge and is the sum of the degrees of all neighbours of vertex u in G. In other words, , Similarly for.

Case (1) If: In we get edges of the type, and. Edges of the type, and are colored in red, green and black respectively as shown in the figure [1] . The number of edges of these types are given in the Table 2.

We know that

i.e.,

Figure 5..

Table 2. Edge partition based on degree sum of neighbors of end vertices of each edge.

Case (2) If: In we get edges of the type and. The number of edges of these types are given in the Table 3.

We know that

i.e.,

Theorem 2.6. The fifth Geometric-arithmetic index () of Dutch windmill graph is

Proof. We know that

Case (1) If: [From Table

2 and Figure 5]

Case (2) If:

Table 3. Edge partition based on degree sum of neighbors of end vertices of each edge.

[From Table 3]

. □

3. Conclusion

The problem of finding the general formula for ABC index, index, Randic connectivity index, Sum connectivity index, GA index and index of Dutch Windmill Graph is solved here analytically without using computers.

Acknowledgements

The first author is also thankful to the University Grants Commission, Government of India for the financial support under the grant MRP(S)-0535/13-14/KAMY004/UGC-SWRO.

Conflict of Interests

The authors declare that there are no conflicts of interests regarding the publication of this paper.

Cite this paper

Kanna, M. , Kumar, R. and Jagadeesh, R. (2016) Computation of Topological Indices of Dutch Windmill Graph.*Open Journal of Discrete Mathematics*, **6**, 74-81. doi: 10.4236/ojdm.2016.62007.

Kanna, M. , Kumar, R. and Jagadeesh, R. (2016) Computation of Topological Indices of Dutch Windmill Graph.

References

[1] Estrada, E., Torres, L., Rodriguez, L. and Gutman, I. (1998) An Atom-Bond Connectivity Index: Modelling the Enthalpy of Formation of Alkanes. Indian Journal of Chemistry, 37A, 849-855.

[2] Chen, J., Liu, J. and Guo, X. (2012) Some Upper Bounds for the Atom-Bond Connectivity Index of Graphs. Applied Mathematics Letters, 25, 1077-1081.

http://dx.doi.org/10.1016/j.aml.2012.03.021

[3] Chen, J. and Guo, X. (2012) The Atom-Bond Connectivity Index of Chemical Bicyclic Graphs. Applied Mathematics— A Journal of Chinese Universities, 27, 243-252.

http://dx.doi.org/10.1007/s11766-012-2756-4

[4] Xing, R., Zhou, B. and Dong, F. (2011) On Atom-Bond Connectivity Index of Connected Graphs. Discrete Applied Mathematics, 159, 1617-1630.

http://dx.doi.org/10.1016/j.dam.2011.06.004

[5] Furtula, B., Gravoc, A. and Vukicevic, D. (2009) Atom-Bond Connectivity Index of Trees. Discrete Applied Mathematics, 157, 2828-2835.

http://dx.doi.org/10.1016/j.dam.2009.03.004

[6] Gutman, I., Furtula, B. and Ivanovic, M. (2012) Notes on Trees with Minimal Atom-Bond Connectivity Index. MATCH Communications in Mathematical and in Computer Chemistry, 67, 467-482.

[7] Xing, R., Zhou, B. and Du, Z. (2010) Further Results on Atom-Bond Connectivity Index of Trees. Discrete Applied Mathematics, 157, 1536-1545.

http://dx.doi.org/10.1016/j.dam.2010.05.015

[8] Xing, R. and Zhou, B. (2012) Extremal Trees with Fixed Degree Sequence for Atom-Bond Connectivity Index. FILOMAT, 26, 683-688.

http://dx.doi.org/10.2298/FIL1204683X

[9] Ghorbani, M. and Hosseinzadeh, M.A. (2010) Computing ABC4 Index of Nanostar Dendrimers. Optoelectronics and Advanced Materials: Rapid Communications, 4, 1419-1422.

[10] Farahani, M.R. (2013) Computing Fourth Atom-Bond Connectivity Index of V-Phenylenic Nanotubes and Nanotori. Acta Chimica Slovenica, 60, 429-432.

[11] Farahani, M.R. (2013) On the Fourth Atom-Bond Connectivity Index of Armchair Polyhex Nanotube. Proceedings of the Romanian Academy—Series B, 15, 3-6.

[12] Randic, M. (1975) On Characterization of Molecular Branching. Journal of the American Chemical Society, 97, 6609-6615.

http://dx.doi.org/10.1021/ja00856a001

[13] Zhou, B. and Xing, R. (2011) On Atom-Bond Connectivity Index. Zeitschrift für Naturforschung, 66a, 61-66.

http://dx.doi.org/10.5560/ZNA.2011.66a0061

[14] Zhou, B. and Trinajstic, N. (2009) On a Novel Connectivity Index. Journal of Mathematical Chemistry, 46, 1252-1270.

http://dx.doi.org/10.1007/s10910-008-9515-z

[15] Zhou, B. and Trinajstic, N. (2010) On General Sum-Connectivity Index. Journal of Mathematical Chemistry, 47, 210-218.

http://dx.doi.org/10.1007/s10910-009-9542-4

[16] Vukicevic, D. and Furtula, B. (2009) Topological Index Based on the Ratios of Geometrical and Arithmetical Means of End-Vertex Degrees of Edges. Journal of Mathematical Chemistry, 46, 1369-1376.

http://dx.doi.org/10.1007/s10910-009-9520-x

[17] Chen, S. and Liu, W. (2010) The Geometric-Arithemtic Index of Nanotubes. Journal of Computational and Theoretical Nanoscience, 7, 1993-1995.

http://dx.doi.org/10.1166/jctn.2010.1573

[18] Das, K.C. and Trinajstic, N. (2010) Comparision between First Geometric-Arithmetic Index and Atom-Bond Connectivity Index. Chemical Physics Letters, 497, 149-151.

http://dx.doi.org/10.1016/j.cplett.2010.07.097

[19] Xiao, L., Chen, S., Guo, Z. and Chen, Q. (2010) The Geometric-Arithmetic Index of Benzenoidsystems and Phenylenes. International Journal of Contemporary Mathematical Sciences, 5, 2225-2230.

[20] Graovac, A.. Ghorbani, M. and Hosseinzadeh, M.A. (2011) Computing Fifth Geometric-Arithmetic Index for Nanostar Dendrimers. Journal of Mahematical Nanoscience, 1, 33-42.

[1] Estrada, E., Torres, L., Rodriguez, L. and Gutman, I. (1998) An Atom-Bond Connectivity Index: Modelling the Enthalpy of Formation of Alkanes. Indian Journal of Chemistry, 37A, 849-855.

[2] Chen, J., Liu, J. and Guo, X. (2012) Some Upper Bounds for the Atom-Bond Connectivity Index of Graphs. Applied Mathematics Letters, 25, 1077-1081.

http://dx.doi.org/10.1016/j.aml.2012.03.021

[3] Chen, J. and Guo, X. (2012) The Atom-Bond Connectivity Index of Chemical Bicyclic Graphs. Applied Mathematics— A Journal of Chinese Universities, 27, 243-252.

http://dx.doi.org/10.1007/s11766-012-2756-4

[4] Xing, R., Zhou, B. and Dong, F. (2011) On Atom-Bond Connectivity Index of Connected Graphs. Discrete Applied Mathematics, 159, 1617-1630.

http://dx.doi.org/10.1016/j.dam.2011.06.004

[5] Furtula, B., Gravoc, A. and Vukicevic, D. (2009) Atom-Bond Connectivity Index of Trees. Discrete Applied Mathematics, 157, 2828-2835.

http://dx.doi.org/10.1016/j.dam.2009.03.004

[6] Gutman, I., Furtula, B. and Ivanovic, M. (2012) Notes on Trees with Minimal Atom-Bond Connectivity Index. MATCH Communications in Mathematical and in Computer Chemistry, 67, 467-482.

[7] Xing, R., Zhou, B. and Du, Z. (2010) Further Results on Atom-Bond Connectivity Index of Trees. Discrete Applied Mathematics, 157, 1536-1545.

http://dx.doi.org/10.1016/j.dam.2010.05.015

[8] Xing, R. and Zhou, B. (2012) Extremal Trees with Fixed Degree Sequence for Atom-Bond Connectivity Index. FILOMAT, 26, 683-688.

http://dx.doi.org/10.2298/FIL1204683X

[9] Ghorbani, M. and Hosseinzadeh, M.A. (2010) Computing ABC4 Index of Nanostar Dendrimers. Optoelectronics and Advanced Materials: Rapid Communications, 4, 1419-1422.

[10] Farahani, M.R. (2013) Computing Fourth Atom-Bond Connectivity Index of V-Phenylenic Nanotubes and Nanotori. Acta Chimica Slovenica, 60, 429-432.

[11] Farahani, M.R. (2013) On the Fourth Atom-Bond Connectivity Index of Armchair Polyhex Nanotube. Proceedings of the Romanian Academy—Series B, 15, 3-6.

[12] Randic, M. (1975) On Characterization of Molecular Branching. Journal of the American Chemical Society, 97, 6609-6615.

http://dx.doi.org/10.1021/ja00856a001

[13] Zhou, B. and Xing, R. (2011) On Atom-Bond Connectivity Index. Zeitschrift für Naturforschung, 66a, 61-66.

http://dx.doi.org/10.5560/ZNA.2011.66a0061

[14] Zhou, B. and Trinajstic, N. (2009) On a Novel Connectivity Index. Journal of Mathematical Chemistry, 46, 1252-1270.

http://dx.doi.org/10.1007/s10910-008-9515-z

[15] Zhou, B. and Trinajstic, N. (2010) On General Sum-Connectivity Index. Journal of Mathematical Chemistry, 47, 210-218.

http://dx.doi.org/10.1007/s10910-009-9542-4

[16] Vukicevic, D. and Furtula, B. (2009) Topological Index Based on the Ratios of Geometrical and Arithmetical Means of End-Vertex Degrees of Edges. Journal of Mathematical Chemistry, 46, 1369-1376.

http://dx.doi.org/10.1007/s10910-009-9520-x

[17] Chen, S. and Liu, W. (2010) The Geometric-Arithemtic Index of Nanotubes. Journal of Computational and Theoretical Nanoscience, 7, 1993-1995.

http://dx.doi.org/10.1166/jctn.2010.1573

[18] Das, K.C. and Trinajstic, N. (2010) Comparision between First Geometric-Arithmetic Index and Atom-Bond Connectivity Index. Chemical Physics Letters, 497, 149-151.

http://dx.doi.org/10.1016/j.cplett.2010.07.097

[19] Xiao, L., Chen, S., Guo, Z. and Chen, Q. (2010) The Geometric-Arithmetic Index of Benzenoidsystems and Phenylenes. International Journal of Contemporary Mathematical Sciences, 5, 2225-2230.

[20] Graovac, A.. Ghorbani, M. and Hosseinzadeh, M.A. (2011) Computing Fifth Geometric-Arithmetic Index for Nanostar Dendrimers. Journal of Mahematical Nanoscience, 1, 33-42.