Schultz Polynomials and Their Topological Indices of Jahangir Graphs *J*_{2,m}

Author(s)
Shaohui Wang^{1,2},
Mohammad Reza Farahani^{3}^{*},
M. R. Rajesh Kanna^{4},
R. Pradeep Kumar^{5}

Affiliation(s)

^{1}
Department of Mathematics, University of Mississippi, Oxford, USA.

^{2}
Department of Mathematics and Computer Science, Adelphi University, Garden City, USA.

^{3}
Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, Tehran, Iran.

^{4}
Department of Mathematics, Maharani's Science College for Women, Mysore, India.

^{5}
Department of Mathematics, The National Institute of Engineering, Mysuru, India.

ABSTRACT

Let G = (V; E) be a simple connected graph. The Wiener index is the sum of distances between all pairs of vertices of a connected graph. The Schultz topological index is equal to and the Modified Schultz topological index is . In this paper, the Schultz, Modified Schultz polynomials and their topological indices of Jahangir graphs*J*_{2,m} for all
integer number m ≥ 3 are calculated.

Let G = (V; E) be a simple connected graph. The Wiener index is the sum of distances between all pairs of vertices of a connected graph. The Schultz topological index is equal to and the Modified Schultz topological index is . In this paper, the Schultz, Modified Schultz polynomials and their topological indices of Jahangir graphs

Received 20 June 2016; accepted 26 August 2016; published 29 August 2016

1. Introduction

Let G = (V; E) be an undirected connected graph without loops or multiple edges. The sets of vertices and edges of G are denoted by V(G) and E(G), respectively. A topological index is a numerical quantity derived in an unambiguous manner from the structure graph of a molecule. As a graph structural invariant, i.e. it does not depend on the labelling or the pictorial representation of a graph. Various topological indices usually reflect molecular size and shape. An oldest topological index in chemistry is the Wiener index, that first introduced by Harold Wiener in 1947 to study the boiling points of paraffin. It plays an important role in the so-called inverse structure-property relationship problems. The Wiener index of a molecular graph G was defined as [1] :

(1)

In 1989, H.P. Schultz [13] has introduced a graph theoretical descriptor for characterizing alkanes by an integer number as follow:

(2)

where d_{u} and d_{v} are degrees of vertices u and v. Schultz named this descriptor the “molecular topological index” and denoted it by MTI. Later MTI became much better known under the name the Schultz index.

In 1997, S. Klavžar and I. Gutman [14] defined another based structure descriptors the Modified Schultz index of G is defined as:

(3)

Now, there are two topological polynomials of a graph G as follow:

(4)

and

^{ }(5)

For more details about the Schultz, Modified Schultz polynomials and their topological indices and other molecular topological polynomials and indices reader can see the paper series [13] - [29] .

In this paper we study the Schultz, Modified Schultz polynomials and their topological indices of Jahangir graphs J_{2,m }for all integer number m ≥ 3.

2. Main Results

In this section we compute the Schultz, Modified Schultz polynomials and their topological indices for Jahangir graphs J_{2,m }"m ≥ 3. The general form of Jahangir graphs J_{n}_{,m} is defined as follows:

Definition 1. [30] - [35] Jahangir graphs J_{n}_{,m }for m ≥ 3, is a graph on nm + 1 vertices i.e., a graph consisting of a cycle C_{nm} with one additional vertex which is adjacent to m vertices of C_{nm} at distance n to each other on C_{nm}.

Theorem 1. Let J_{2,m} be the Jahangir graphs ("m ≥ 3). Then,

The Schultz polynomial of J_{2,m} is equal to

The Modified Schultz polynomial of J_{2,m} is equal to

Proof. "m ≥ 3 consider Jahangir graph J_{2,m}. By using Definition 1 and [29] - [32] , one can see that the number

of vertices in Jahangir graph J_{2,m} is equal to And the number of edges of Ja-

hangir graph J_{2,m} is equal to Because, there is only Center vertex with

degree m and there are m vertices with degree 2 and m vertices with degree. In this paper, we denote the sets of all vertices with degree two by A, all vertices with degree three by B and only Center vertex c by C.

From the structure of Jahangir graph J_{2,m} (Figure 1), we see that there are distances from one to four, for every vertices In other words, and the Diameter D of Jahangir graph J_{2,m} is equal to D(J_{2,m}) = 4.

I. If, , we have two case for first sentences of the Schultz, Modified Schultz polynomials of J_{2,m}.

I-1. For a vertex, there are two path with length one until a vertex, thus there are 2m edges uvÎE(J_{2,m}), such that,. Therefore, we have two terms 5 × 2mx^{1}, 6 × 2mx^{1} of the Schultz and Modified Schultz polynomials of Jahangir graph J_{2,m}, respectively.

I-2. For only vertex, there are m path with length one until a vertex, thus there are m edges, such that,. So, we have two terms and ^{ }of the Schultz and Modified Schultz polynomials of J_{2,m}, respectively.

Thus, the first sentences of the Schultz and Modified Schultz polynomials of Jahangir graph J_{2,m} are equal to

and, respectively.

II. If, , we have three case for first sentences of the Schultz, Modified Schultz polynomials of J_{2,m}.

II-1. For a vertex, there are two path with length two until other vertices A, so there are (1/2) × 2m 2-edge-path in J_{2,m}, such that. Therefore, we have a terms 4 × mx^{2 }of the Schultz and Modified Schultz polynomials of J_{2,m}.

II-2. For every vertex, there are only 2-edge-path until the Center vertex c, and there are m

2-edge-path in J_{2,m }with and. Therefore, we have two terms, 2m × mx^{2} of the Schultz and Modified Schultz polynomials of J_{2,m}, respectively.

II-3. For a vertex, there are m − 1 path with length two until other vertices, so there are 2-edge-path in J_{2,m}, such that,. So, we have two terms and of the Schultz and Modified Schultz polynomials of J_{2,m}, respectively.

Thus, the second sentences of the Schultz and Modified Schultz polynomials of Jahangir graph J_{2,m} are equal

to and respectively.

III. If, , for a vertex, there are (m − 2)m path with length three until vertices of B, such that,. Therefore, we have two sentences and of the Schultz and Modified Schultz polynomials of Jahangir graph J_{2,m}, respectively.

IV. If, , for a vertex, there are m − 3 path with length 4 = D(J_{2,m}), between v and other vertices u of A. Thus by, the fourth sentence of the Schultz and Modified Schultz polynomials of Jahangir graph J_{2,m} is equal to.

From the definition of the Schultz, Modified Schultz polynomials and above mentions, we have following results "m Î ℕ − {2}.

Figure 1. Jahangir graphs J_{2,4,} J_{2,5,} J_{2,6,} J_{2,16 }and J_{2,32} [32] .

(6)

and

(7)

And these complete the proof.

Theorem 2. Let J_{2,m} be the Jahangir graphs ("m ≥ 3). Then, the Schultz, Modified Schultz indices of J_{2,m} are equal to

Proof. Consider the Jahangir graph J_{2,m} ("m ≥ 3) that presented in above proof. Now, by using the results from proof of Theorem 1 and according to the definitions of the Schultz, Modified Schultz indices of the graph G, one can see that these indices are the first derivative of their polynomials (evaluated at x = 1). Thus we have following computations "m Î ℕ − {2}.

(8)

And

(9)

Here the proof of theorem is completed.

Acknowledgements

The author is thankful to Professor Emeric Deutsch from Department of Mathematics of Polytechnic University (Brooklyn, NY 11201, USA) for his precious support and suggestions.

NOTES

^{*}Corresponding author.

Cite this paper

Wang, S. , Farahani, M. , Kanna, M. , Kumar, R. (2016) Schultz Polynomials and Their Topological Indices of Jahangir Graphs*J*_{2,m}. *Applied Mathematics*, **7**, 1632-1637. doi: 10.4236/am.2016.714140.

Wang, S. , Farahani, M. , Kanna, M. , Kumar, R. (2016) Schultz Polynomials and Their Topological Indices of Jahangir Graphs

References

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http://dx.doi.org/10.1021/ja01193a005

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

[3] Dobrynin, A.A., Entringer, R. and Gutman, I. (2001) Wiener Index of Trees: Theory and Applications. Acta Applicandae Mathematica, 66, 211-249.

http://dx.doi.org/10.1023/A:1010767517079

[4] Knor, M., Poto?nik, P. and ?krekovski, R. (2013) Wiener Index of Iterated Line Graphs of Trees Homeomorphic to the Claw K1;3. Ars Mathematica Contemporanea, 6, 211-219.

[5] Gutman, I. and Polansky, O.E. (1986) Mathematical Concepts in Organic Chemistry. Springer, Berlin.

http://dx.doi.org/10.1007/978-3-642-70982-1

[6] Gutman, I., Klav?ar, S. and Mohar, B. (1997) Fifty Years of the Wiener Index. MATCH Commun. Math. Comput. Chem., 35, 1-259.

[7] Gutman, I., Klav?ar, S. and Mohar, B. (1997) Fiftieth Anniversary of the Wiener Index. Discrete Appl. Math., 80, 1-113.

[8] Farahani, M.R. (2013) Hosoya Polynomial, Wiener and Hyper-Wiener Indices of Some Regular Graphs. Informatics Engineering, an International Journal (IEIJ), 1, 9-13.

[9] Farahani, M.R. (2013) Computing Hosoya Polynomial, Wiener Index and Hyper-Wiener Index of Harary Graph. Iranian Journal of Mathematical Chemistry, 4, 235-240.

[10] Farahani, M.R. and Vlad, M.P. (2012) On the Schultz, Modified Schultz and Hosoya Polynomials and Derived Indices of Capra-Designed Planar Benzenoid. Studia UBB Chemia, 57, 55-63.

[11] Farahani, M.R. (2013) Hosoya, Schultz, Modified Schultz Polynomials and Their Topological Indices of Benzene Molecules: First Members of Polycyclic Aromatic Hydrocarbons (PAHs). International Journal of Theoretical Chemistry, 1, 9-16.

[12] Farahani, M.R. (2013) On the Schultz Polynomial, Modified Schultz Polynomial, Hosoya Polynomial and Wiener Index of Circumcoronene Series of Benzenoid. Journal of Applied Mathematics & Informatics, 31, 595-608.

http://dx.doi.org/10.14317/jami.2013.595

[13] Schultz, H.P. (1989) Topological Organic Chemistry 1. Graph Theory and Topologicalindices of Alkanes. Journal of Chemical Information and Computer Science, 29, 227-228.

http://dx.doi.org/10.1021/ci00063a012

[14] Klav?ar, S. and Gutman, I. (1996) A Comparison of the Schultz Molecular Topological Index with the Wiener Index. Journal of Chemical Information and Computer Science, 36, 1001-1003.

http://dx.doi.org/10.1021/ci9603689

[15] Devillers, J. and Balaban, A.T. (1999) Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam.

[16] Todeschini, R. and Consonni, V. (2000) Handbook of Molecular Descriptors. Wiley-VCH, Weinheim.

http://dx.doi.org/10.1002/9783527613106

[17] Karelson, M. (2000) Molecular Descriptors in QSAR/QSPR. Wiley Interscience, New York.

[18] Iranmanesh, A. and Alizadeh, Y. (2008) Computing Wiener and Schultz Indices of HAC5C7[p,q] Nanotube by GAP Program. American Journal of Applied Sciences, 5, 1754-1757.

http://dx.doi.org/10.3844/ajassp.2008.1754.1757

[19] Eliasi, M. and Taeri, B. (2008) Schultz Polynomials of Composite Graphs. Applicable Analysis and Discrete Mathematics, 2, 285-296.

http://dx.doi.org/10.2298/AADM0802285E

[20] Iranmanesh, A. and Alizadeh, Y. (2009) Computing Szeged and Schultz Indices of HAC5C7C9[p,q] Nanotube By Gap Program. Digest Journal of Nanomaterials and Biostructures, 4, 67-72.

[21] Alizadeh, Y., Iranmanesh, A. and Mirzaie, S. (2009) Computing Schultz Polynomial, Schultz Index of C60 Fullerene by Gap Program. Digest Journal of Nanomaterials and Biostructures, 4, 7-10.

[22] Iranmanesh, A. and Alizadeh, Y. (2009) Computing Hyper-Wiener and Schultz Indices of TUZC6[p;q] Nanotube by Gap Program. Digest Journal of Nanomaterials and Biostructures, 4, 607-611.

[23] Halakoo, O., Khormali, O. and Mahmiani, A. (2009) Bounds for Schultz Index of Pentachains. Digest Journal of Nanomaterials and Biostructures, 4, 687-691.

[24] Heydari, A. (2010) On the Modified Schultz Index of C4C8(S) Nanotubes and Nanororus. Digest Journal of Nanomaterials and Biostructures, 5, 51-56.

[25] Hedyari, A. (2011) Wiener and Schultz Indices of V-Naphtalenic Nanotori. Optoelectronics and Advanced Materials— Rapid Communications, 5, 786-789.

[26] Farahani, M.R. (2013) On the Schultz and Modified Schultz Polynomials of Some Harary Graphs. International Journal of Applications of Discrete Mathematics, 1, 1-8.

[27] Wang, S. and Wei, B. (2016) Multiplicative Zagreb Indices of Cactus Graphs. Discrete Mathematics, Algorithms and Applications, 8, 1650040.

http://dx.doi.org/10.1142/S1793830916500403

[28] Wang, C., Wang, S. and Wei, B. (2016) Cacti with Extremal PI Index. Transactions on Combinatorics, 5, 1-8.

[29] Wang, S. and Wei, B. (2015) Multiplicative Zagreb Indices of K-Trees. Discrete Applied Mathematics, 180, 168-175.

http://dx.doi.org/10.1016/j.dam.2014.08.017

[30] Ali, K., Baskoro, E.T. and Tomescu, I. (2008) On the Ramzey Number of Paths and Jahangir Graph J3,m. Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, 51, 177-182.

[31] Mojdeh, D.A. and Ghameshlou, A.N. (2007) Domination in Jahangir Graph J2,m. International Journal of Contemporary Mathematical Sciences, 2, 1193-1199.

http://dx.doi.org/10.12988/ijcms.2007.07122

[32] Farahani, M.R. (2015) Hosoya Polynomial and Wiener Index of Jahangir Graphs J2,m. Pacific Journal of Applied Mathematics, 7. (In Press)

[33] Farahani, M.R. (2015) The Wiener Index and Hosoya Polynomial of a Class of Jahangir Graphs J3,m. Fundamental Journal of Mathematics and Mathematical Science, 3, 91-96.

[34] Farahani, M.R. (2015) Hosoya Polynomial of Jahangir Graphs J4,m. Global Journal of Mathematics, 3, 232-236.

[35] Wang, S., Farahani, M.R., Rajesh Kanna, M.R., Jamil, M.K. and Kumar, P.R. (2016) Wiener index and Hosoya polynomial of Jahangir graphs J5,m. Applied and Computational Mathematics, 5, 138-141.

[1] Wiener, H. (1947) Structural Determination of Paraffin Boiling Points. Journal of the American Chemical Society, 69, 17-20.

http://dx.doi.org/10.1021/ja01193a005

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

[3] Dobrynin, A.A., Entringer, R. and Gutman, I. (2001) Wiener Index of Trees: Theory and Applications. Acta Applicandae Mathematica, 66, 211-249.

http://dx.doi.org/10.1023/A:1010767517079

[4] Knor, M., Poto?nik, P. and ?krekovski, R. (2013) Wiener Index of Iterated Line Graphs of Trees Homeomorphic to the Claw K1;3. Ars Mathematica Contemporanea, 6, 211-219.

[5] Gutman, I. and Polansky, O.E. (1986) Mathematical Concepts in Organic Chemistry. Springer, Berlin.

http://dx.doi.org/10.1007/978-3-642-70982-1

[6] Gutman, I., Klav?ar, S. and Mohar, B. (1997) Fifty Years of the Wiener Index. MATCH Commun. Math. Comput. Chem., 35, 1-259.

[7] Gutman, I., Klav?ar, S. and Mohar, B. (1997) Fiftieth Anniversary of the Wiener Index. Discrete Appl. Math., 80, 1-113.

[8] Farahani, M.R. (2013) Hosoya Polynomial, Wiener and Hyper-Wiener Indices of Some Regular Graphs. Informatics Engineering, an International Journal (IEIJ), 1, 9-13.

[9] Farahani, M.R. (2013) Computing Hosoya Polynomial, Wiener Index and Hyper-Wiener Index of Harary Graph. Iranian Journal of Mathematical Chemistry, 4, 235-240.

[10] Farahani, M.R. and Vlad, M.P. (2012) On the Schultz, Modified Schultz and Hosoya Polynomials and Derived Indices of Capra-Designed Planar Benzenoid. Studia UBB Chemia, 57, 55-63.

[11] Farahani, M.R. (2013) Hosoya, Schultz, Modified Schultz Polynomials and Their Topological Indices of Benzene Molecules: First Members of Polycyclic Aromatic Hydrocarbons (PAHs). International Journal of Theoretical Chemistry, 1, 9-16.

[12] Farahani, M.R. (2013) On the Schultz Polynomial, Modified Schultz Polynomial, Hosoya Polynomial and Wiener Index of Circumcoronene Series of Benzenoid. Journal of Applied Mathematics & Informatics, 31, 595-608.

http://dx.doi.org/10.14317/jami.2013.595

[13] Schultz, H.P. (1989) Topological Organic Chemistry 1. Graph Theory and Topologicalindices of Alkanes. Journal of Chemical Information and Computer Science, 29, 227-228.

http://dx.doi.org/10.1021/ci00063a012

[14] Klav?ar, S. and Gutman, I. (1996) A Comparison of the Schultz Molecular Topological Index with the Wiener Index. Journal of Chemical Information and Computer Science, 36, 1001-1003.

http://dx.doi.org/10.1021/ci9603689

[15] Devillers, J. and Balaban, A.T. (1999) Topological Indices and Related Descriptors in QSAR and QSPR. Gordon and Breach, Amsterdam.

[16] Todeschini, R. and Consonni, V. (2000) Handbook of Molecular Descriptors. Wiley-VCH, Weinheim.

http://dx.doi.org/10.1002/9783527613106

[17] Karelson, M. (2000) Molecular Descriptors in QSAR/QSPR. Wiley Interscience, New York.

[18] Iranmanesh, A. and Alizadeh, Y. (2008) Computing Wiener and Schultz Indices of HAC5C7[p,q] Nanotube by GAP Program. American Journal of Applied Sciences, 5, 1754-1757.

http://dx.doi.org/10.3844/ajassp.2008.1754.1757

[19] Eliasi, M. and Taeri, B. (2008) Schultz Polynomials of Composite Graphs. Applicable Analysis and Discrete Mathematics, 2, 285-296.

http://dx.doi.org/10.2298/AADM0802285E

[20] Iranmanesh, A. and Alizadeh, Y. (2009) Computing Szeged and Schultz Indices of HAC5C7C9[p,q] Nanotube By Gap Program. Digest Journal of Nanomaterials and Biostructures, 4, 67-72.

[21] Alizadeh, Y., Iranmanesh, A. and Mirzaie, S. (2009) Computing Schultz Polynomial, Schultz Index of C60 Fullerene by Gap Program. Digest Journal of Nanomaterials and Biostructures, 4, 7-10.

[22] Iranmanesh, A. and Alizadeh, Y. (2009) Computing Hyper-Wiener and Schultz Indices of TUZC6[p;q] Nanotube by Gap Program. Digest Journal of Nanomaterials and Biostructures, 4, 607-611.

[23] Halakoo, O., Khormali, O. and Mahmiani, A. (2009) Bounds for Schultz Index of Pentachains. Digest Journal of Nanomaterials and Biostructures, 4, 687-691.

[24] Heydari, A. (2010) On the Modified Schultz Index of C4C8(S) Nanotubes and Nanororus. Digest Journal of Nanomaterials and Biostructures, 5, 51-56.

[25] Hedyari, A. (2011) Wiener and Schultz Indices of V-Naphtalenic Nanotori. Optoelectronics and Advanced Materials— Rapid Communications, 5, 786-789.

[26] Farahani, M.R. (2013) On the Schultz and Modified Schultz Polynomials of Some Harary Graphs. International Journal of Applications of Discrete Mathematics, 1, 1-8.

[27] Wang, S. and Wei, B. (2016) Multiplicative Zagreb Indices of Cactus Graphs. Discrete Mathematics, Algorithms and Applications, 8, 1650040.

http://dx.doi.org/10.1142/S1793830916500403

[28] Wang, C., Wang, S. and Wei, B. (2016) Cacti with Extremal PI Index. Transactions on Combinatorics, 5, 1-8.

[29] Wang, S. and Wei, B. (2015) Multiplicative Zagreb Indices of K-Trees. Discrete Applied Mathematics, 180, 168-175.

http://dx.doi.org/10.1016/j.dam.2014.08.017

[30] Ali, K., Baskoro, E.T. and Tomescu, I. (2008) On the Ramzey Number of Paths and Jahangir Graph J3,m. Bulletin mathématique de la Société des Sciences Mathématiques de Roumanie, 51, 177-182.

[31] Mojdeh, D.A. and Ghameshlou, A.N. (2007) Domination in Jahangir Graph J2,m. International Journal of Contemporary Mathematical Sciences, 2, 1193-1199.

http://dx.doi.org/10.12988/ijcms.2007.07122

[32] Farahani, M.R. (2015) Hosoya Polynomial and Wiener Index of Jahangir Graphs J2,m. Pacific Journal of Applied Mathematics, 7. (In Press)

[33] Farahani, M.R. (2015) The Wiener Index and Hosoya Polynomial of a Class of Jahangir Graphs J3,m. Fundamental Journal of Mathematics and Mathematical Science, 3, 91-96.

[34] Farahani, M.R. (2015) Hosoya Polynomial of Jahangir Graphs J4,m. Global Journal of Mathematics, 3, 232-236.

[35] Wang, S., Farahani, M.R., Rajesh Kanna, M.R., Jamil, M.K. and Kumar, P.R. (2016) Wiener index and Hosoya polynomial of Jahangir graphs J5,m. Applied and Computational Mathematics, 5, 138-141.