2017
8
1
0
91
Stirling Numbers and Generalized Zagreb Indices
2
2
We show how generalized Zagreb indices $M_1^k(G)$ can be computed by using a simple graph polynomial and Stirling numbers of the second kind. In that way we explain and clarify the meaning of a triangle of numbers used to establish the same result in an earlier reference.
1

1
5


T.
Doslic
1Department of Mathematics, Faculty of Civil Engineering, University of Zagreb,
1Department of Mathematics, Faculty of Civil
Croatia
doslic@grad.hr


S.
Sedghi
Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshar, Iran
Department of Mathematics, Qaemshahr Branch,
I R Iran
sedghi_gh@yahoo.com


N.
Shobe
Department of Mathematics, Babol Branch,
Islamic Azad
University, Babol, Iran
Department of Mathematics, Babol Branch,
I R Iran
nabi_shobe@yahoo.com
Simple Graph
Zagreb index
Stirling number
[1. E. Deutsch, S. Klavžar, M–polynomial and degree–based topological indices,##Iranian J. Math. Chem. 6 (2015) 93–102.##2. T. Došlić, M. Ghorbani, M. A. Hosseinzadeh, Eccentric connectivity polynomial of##some graph operations, Util. Math. 84 (2011) 297–309.##3. G. H. Fath–Tabar, A. Azad, N. Elahinezhad, Some topological indices of tetrameric##1,3–adamantane, Iranian J. Math. Chem. 1 (2010) 111–118.##4. B. Furtula, I. Gutman, A forgotten topological index, J. Math. Chem. 53 (2015)##1184–1190.##5. R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison–Wesley,##Reading, 1988.##6. I. Gutman, K. Ch. Das, The first Zagreb index 30 years after, MATCH Commun.##Math. Comput. Chem. 50 (2004) 83–92.##7. I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total electron##energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972) 535–538.##8. X. Li, J. Zheng, A unified approach to the extremal trees for different indices,##MATCH Commun. Math. Comput. Chem. 54 (2005) 195–208.##9. S. Sedghi, N. Shobe, M. A. Salahshoor, The polynomials of a graph, Iranian J.##Math. Sci. Inf. 3 (2008) 55–68.##10. G. B. A. Xavier, E. Suresh, I. Gutman, Counting relations for general Zagreb##indices, Kragujevac J. Math. 38 (2014) 95–103.##]
Relationship between Coefficients of Characteristic Polynomial and Matching Polynomial of Regular Graphs and its Applications
2
2
ABSTRACT. Suppose G is a graph, A(G) its adjacency matrix and f(G, x)=x^n+a_(n1)x^(n1)+... is the characteristic polynomial of G. The matching polynomial of G is defined as M(G, x) = x^nm(G,1)x^(n2) + ... where m(G,k) is the number of kmatchings in G. In this paper, we determine the relationship between 2kth coefficient of characteristic polynomial, a_(2k), and kth coefficient of matching polynomial, (1)^km(G, k), in a regular graph. In the rest of this paper, we apply these relations for finding 5,6matchings of fullerene graphs.
1

7
23


F.
Taghvaee
University of Kashan
University of Kashan
I R Iran
taghvaei19@yahoo.com


G.
FathTabar
University of Kashan
University of Kashan
I R Iran
fathtabar@kashanu.ac.ir
Characteristic polynomial
Matching polynomial
Fullerene graph
[1. A. R. Ashrafi and G. H. FathTabar, Bounds on the Estrada index of ISR (4, 6)–##fullerenes, Appl. Math. Lett. 24 (2011) 337–339.##2. A. Behmaram, Matching in Fullerene and Molecular Graphs, Ph.D. thesis,##University of Tehran, 2013.##3. A. Behmaram, T. Došlić and S. Friedland, Matchings in m–generalized fullerene##graphs, Ars Math. Contemp. 11 (2016) 301–313.##4. A. Behmaram, H. Yousefi Azari and A. R. Ashrafi, Closed formulas for the number##of small paths, independent sets and matchings in fullerenes, Appl. Math. Lett. 25##(2012) 1721–1724.##5. A. Behmaram, On the number of 4matchings in graphs, MATCH Commun. Math.##Comput. Chem. 62 (2009) 381–388.##6. N. Biggs, Algebraic Graph Theory, Cambridge Univ. Press, Cambridge, 1974.##7. D. M. Cvetković, M. Doob and H. Sachs, Spectra of Graphs: Theory and##Applications, Academic Press, New York, 1980.##8. M. Deza, M. Dutour and P. W. Fowler, Zigzags, railroads and knots in fullerenes,##J. Chem. Inf. Comp. Sci. 44 (2004) 1282–1293.##9. E. J. Farrell, An introduction to matching polynomials, J. Combin. Theory Ser. B 27##(1979) 75–86. ##10. G. H. FathTabar, A. R. Ashrafi and I. Gutman, Note on Estrada and LEstrada##indices of graphs, Bull. Cl. Sci. Math. Nat. Sci. Math. 34 (2009) 1–16.##11. G. H. FathTabar, A. R. Ashrafi and D. Stevanović, Spectral properties of##fullerenes, J. Comput. Theor. Nanosci. 9 (2012) 327–329.##12. P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Oxford Univ. Press,##Oxford, 1995.##13. M. Ghorbani and E. BaniAsadi, Remarks on characteristic coefficients of##fullerene graphs, Appl. Math. Comput. 230 (2014) 428–435.##14. H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl and R. E. Smalley, C60:##buckminsterfullerene, Nature 318 (1985) 162–163.##15. Z. Mehranian, A. Gholami and A. R. Ashrafi, Experimental results on the symmetry##and topology of 3 and 4generalized fullerenes, J. Comput. Theor. Nanosci. 11##(2014) 1–6.##16. W. Myrvold, B. Bultena, S. Daugherty, B. Debroni, S. Girn, M. Minchenko, J.##Woodcock and P. W. Fowler, FuiGui: A graphical user interface for investigating##conjectures about fullerenes, MATCH Commun. Math. Comput. Chem. 58 (2007)##403–422.##17. P. Schwerdtfeger, L. Wirz and J. Avery, Program fullerene: a software package for##constructing and analyzing structures of regular fullerenes, J. Comput. Chem. 34##(2013) 1508–1526.##18. M. D. Sikirić and M. Deza, Space fullerenes; computer search for new##FrankKasper structures II, Structural Chemistry, 23 (2012) 1103–1114.##19. M. D. Sikirić, O. DelgadoFriedrichs and M. Deza, Space fullerenes: a computer##search for new Frank–Kasper structures, Acta Crystallogr. A 66 (2010) 602–615.##20. F. Taghvaee and A. R. Ashrafi, Ordering some regular graphs with respect to##spectral moments, submitted.##21. F. Taghvaee and A. R. Ashrafi, On spectrum of Igraphs and its ordering with##respect to spectral moments, submitted.##22. F. Taghvaee and A. R. Ashrafi, Comparing fullerenes by spectral moments, J.##Nanosci. Nanotechnol. 16 (2016) 3132–3135.##23. F. Taghvaee and G. H. FathTabar, Signless Laplacian spectral moments of graphs##and ordering some graphs with respect to them, Alg. Struc. Appl. 1 (2014) 133–141.##24. R. Vesalian and F. Asgari, Number of 5matching in graphs, MATCH Commun.##Math. Comput. Chem. 69 (2013) 33–46. ##]
The Topological Indices of some Dendrimer Graphs
2
2
In this paper the Wiener and hyper Wiener index of two kinds of dendrimer graphs are determined. Using the Wiener index formula, the Szeged, Schultz, PI and Gutman indices of these graphs are also determined.
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25
35


M.
Darafsheh
School of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Tehran, Iran
School of Mathematics, Statistics and Computer
I R Iran


M.
Namdari
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran
Department of Mathematics, Shahid Chamran
I R Iran


S.
Shokrolahi
Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran.
Department of Mathematics, Shahid Chamran
I R Iran
shokrolahisara@yahoo.com
topological index
Dendrimer
Wiener index
Hyper Wiener index
[1. R. Entringer, Distance in graphs: Trees, J. Combin. Math. Combin. Comput. 24##(1997) 65–84.##2. I. Gutman, S. Klavžar, B. Mohar (Eds), Fifty years of the Wiener index, MATCH##Commun. Math. Comput. Chem. 35 (1997) 1–259.##3. H. Hosoya, Topological Index, A Newly Proposed Quantity Characterizing the##Topological Nature of Structural Isomers of Saturated Hydrocarbons, Bull. Chem.##Soc. Jpn. 44 (1971) 2332–2339.##4. H. Wiener, Structural Determination of Paraffin Boiling Points, J. Am. Chem. Soc.##69 (1947) 17–20.##5. I. Gutman, A formula for the Wiener number of trees and its extension to graphs##containing cycles, Graph Theory Notes N.Y. 27 (1994) 9–15.##6. S. Klavžar, A. Rajapakse, I. Gutman, The Szeged and the Wiener index of graphs,##Appl. Math. Lett. 9 (1996) 45–49.##7. P. V. Khadikar, On a Novel structural descriptor PI, Nat. Acad. Sci. Lett. 23 (2000)##113–118.##8. P. V. Khadikar. S. Karmakar and V. K. Agrawal, Relationship and relative##correction potential of the Wiener, Szeged and PI Indices, Nat. Acad. Sci. Lett. 23##(2000) 165–170.##9. H. P. Schultz, T. P. Schultz, Topological organic chemistry. 3. Graph theory, Binary##and Decimal Adjacency Matrices, and Topological indices of Alkanes, J. Chem. Inf.##Comput. Sci. 31 (1991) 144–147.##10. H. P. Schultz, T. P. Schultz, Topological organic chemistry. 6. Theory and##topological indices of cycloalkanes, J. Chem. Inf. Comput. Sci. 33 (1993) 240–244.##11. I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem.##Inf. Comput. Sci. 34 (1994) 1087–1089.##12. I. Gutman, Y. N. Yeh, S. L. Lee, Y. L. Luo, Some recent results in the theory of the##Wiener number, Indian J. Chem. 32 (1993) 651–661.##13. M. Randić, Novel molecular descriptor for structure–property studies, Chem. Phys.##Lett. 211 (1993) 478–483.##14. D. J. Klein, I. Lukovits, I. Gutman, On the definition of the hyper–Wiener index for##cyclecontaining structures, J. Chem. Inf. Comput. Sci. 35 (1995) 5052.##]
On the Multiplicative Zagreb Indices of Bucket Recursive Trees
2
2
Bucket recursive trees are an interesting and natural generalization of ordinary recursive trees and have a connection to mathematical chemistry. In this paper, we give the lower and upper bounds for the moment generating function and moments of the multiplicative Zagreb indices in a randomly chosen bucket recursive tree of size $n$ with maximal bucket size $bgeq1$. Also, we consider the ratio of the multiplicative Zagreb indices for different values of $n$ and $b$. All our results reduce to the ordinary recursive trees for $b=1$.
1

37
45


R.
Kazemi
Imam Khomeini international university
Imam Khomeini international university
I R Iran
r.kazemi@sci.ikiu.ac.ir
Bucket recursive trees
Multiplicative Zagreb index
Moment generating function
Moments
[1. P. Billingsley, Probability and Measure, John Wiley and Sons, New York, 1995.##2. M. Eliasi, A. Iranmanesh, I. Gutman, Multiplicative versions of first Zagreb index,##MATCH Commun. Math. Comput. Chem. 68 (2012), 217–230.##3. A. Iranmanesh, M. A. Hosseinzadeh, I. Gutman, On multiplicative Zagreb indices##of graphs, Iranian J. Math. Chem. 3 (2012), 145–154.##4. R. Kazemi, Probabilistic analysis of the first Zagreb index, Trans. Comb. 2 (2013),##35–40.##5. R. Kazemi, Depth in bucket recursive trees with variable capacities of buckets, Acta##Math. Sin. Engl. Ser. 30 (2014), 305–310.##6. R. Kazemi, The eccentric connectivity index of bucket recursive trees, Iranian J.##Math. Chem. 5 (2014), 77–83.##7. H. M. Mahmoud, R. T. Smythe, Probabilistic analysis of bucket recursive trees,##Theoret. Comput. Sci. 144 (1995), 221–249.##8. A. Meir, J. W. Moon, On the altitude of nodes in random trees, Canadian J. Math.##30 (1978), 997–1015.##9. R. Todeschini, D. Ballabio, V. Consonni, Novel molecular descriptors based on##functions of new vertex degrees, in: I. Gutman, B. Furtula (Eds.), Novel Molecular##Structure Descriptors Theory and Applications I, Univ. Kragujevac, Kragujevac##(2010), 73–100.##10. R. Todeschini, V. Consonni, New local vertex invariants and molecular descriptors##based on functions of the vertex degrees, MATCH Commun. Math. Comput. Chem.##64 (2010), 359–372.##]
The Conditions of the Violations of Le Chatlier’s Principle in Gas Reactions at Constant T and P
2
2
Le Chatelier's principle is used as a very simple way to predict the effect of a change in conditions on a chemical equilibrium. . However, several studies have reported the violation of this principle, still there is no reported simple mathematical equation to express the exact condition of violation in the gas phase reactions. In this article, we derived a simple equation for the violation of Le Chatelier's principle for the ideal gas reactions at the constant temperature and pressure.
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47
52


M.
Torabi Rad
University of Qom, Qom, Iran
University of Qom, Qom, Iran
I R Iran
morteza.0mtr0@yahoo.com


A.
Abbasi
University of Qom, Qom, Iran
University of Qom, Qom, Iran
I R Iran
a.abbasi@qom.ac.ir
Violation of Le Chatelier
Principle gas reaction
Mixture
Chemical equilibria
Chemical potential moderation
[1. H. Le Chatelier, Sur un énoncé général des lois des équilibres chimiques, Compt.##Rend. 99 (1884) 786–789.##2. D. Cheung, Using thinkaloud protocols to investigate secondary school chemistry##teachers’ misconceptions about chemical equilibrium, Chem. Educ. Res. Pract. 10##(2009) 97–108.##3. D. Cheung, The Adverse Effects of Le Chatelier’s Principle on Teacher##Understanding of Chemical Equilibrium, J. Chem. Educ. 86 (2009) 514–518.##4. J. J.De Heer, The principle of Le Chatelier and Braun, Chem. Educ. 34 (1957)##375–380.##5. J. J. De Heer, Le Chatelier, scientific principle, or "sacred cow"?, Chem. Educ. 35##(1958) 133–136.##6. R. S. Treptow, Le Chatelier's principle: A reexamination and method of graphic##illustration, J. Chem. Educ. 57 (1980) 417–420.##7. J. Gold, V. Gold, Neither Le Chateliers nor a principle, Chem. Brit. 20 (1984)##802–806.##8. K. Posthumus, The application of the van't Hoff−le Chatelier−Braun principle to##chemical equilibria, Rec. Tray. Chim. 52 (1933) 25–35.##9. K. Posthumus, The application of the van't Hoff−le Chatelier−Braun principle to##chemical equilibria. II, Rec. Tray. Claim. 53 (1933) 308–311.##10. J. E. Lacy, Equilibria that shift left upon addition of more reactant, J. Chem. Educ.##82 (2005) 1192–1193.##]
Neighbourly Irregular Derived Graphs
2
2
A connected graph G is said to be neighbourly irregular graph if no two adjacent vertices of G have same degree. In this paper we obtain neighbourly irregular derived graphs such as semitotalpoint graph, k^{tℎ} semitotalpoint graph, semitotalline graph, paraline graph, quasitotal graph and quasivertextotal graph and also neighbourly irregular of some graph products.
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53
60


B.
Basavanagoud
KARNATAK UNIVERSITY DHARWAD
KARNATAK UNIVERSITY DHARWAD
India
b.basavanagoud@gmail.com


S.
Patil
Karnatak University
Karnatak University
India
shreekantpatil949@gmail.com


V. R.
Desai
Karnatak University
Karnatak University
India
veenardesai6f@gmail.com


M.
Tavakoli
Ferdowsi University of Mashhad
Ferdowsi University of Mashhad
I R Iran
m_tavakoli@um.ac.ir


A. R.
Ashrafi
University of Kashan
University of Kashan
I R Iran
ashrafi@kashanu.ac.ir
Neighbourly irregular
Derived graphs
Product graphs
[1. Y. Alavi, G. Chartrand, F. R. K. Chung, P. Erdos, H. L. Graham, O. R. ##Oellermann, Highly irregular graphs, J. Graph Theory 11 (1987) 235–249.##2. S. G. Bhragsam, S. K. Ayyaswamy, Neighbourly irregular graphs, Indian J. Pure##Appl. Math. 35(3) (2004) 389–399.##3. T. Došlić, Vertexweighted Wiener polynomials for composite graphs, Ars Math.##Contemp. 1 (2008) 66–80.##4. F. Harary, Graph Theory, AddisonWesley Publishing Co. Inc., Reading, Mass.,##5. Y. Hou, WC. Shiu, The spectrum of the edge corona of two graphs, Electron. J.##Linear Algebra 20 (2010) 586–594.##6. S. R. Jog, S. P. Hande, I. Gutman, S. B. Bozkurt, Derived graphs of some graphs,##Kragujevac J. Math. 36(2) (2012) 309–314.##7. V. R. Kulli, B. Basavanagoud, On the quasivertextotal graph of a graph, J.##Karnatak Uni. Sci. 42 (1998) 1–7.##8. E. Sampathkumar, S. B. Chikkodimath, Semitotal graphs of a graphI, J.##Karnatak Uni. Sci. 18 (1973) 274–280.##9. D. V. S. Sastry, B. Syam Prasad Raju, Graph equations for line graphs, total##graphs, middle graphs and quasitotal graphs, Discrete Math. 48 (1984) 113–119.##10. M. Tavakoli, F. Rahbarnia, A. R. Ashrafi, Studying the corona product of graphs##under some graph invariants, Trans. Comb. 3(3) (2014) 43–49.##11. Y. N. Yeh, I. Gutman, On the sum of all distances in composite graphs, Discrete##Math. 135 (1994) 359–365.##12. H. B. Walikar, S. B. Halkarni, H. S. Ramane, M. Tavakoli, A. R. Ashrafi, On##neighbourly irregular graphs, Kragujevac J. Math. 39(1) (2015) 31–39.##]
Splice Graphs and their VertexDegreeBased Invariants
2
2
Let G_1 and G_2 be simple connected graphs with disjoint vertex sets V(G_1) and V(G_2), respectively. For given vertices a_1in V(G_1) and a_2in V(G_2), a splice of G_1 and G_2 by vertices a_1 and a_2 is defined by identifying the vertices a_1 and a_2 in the union of G_1 and G_2. In this paper, we present exact formulas for computing some vertexdegreebased graph invariants of splice of graphs.
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61
70


M.
Azari
Islamic Azad University
Islamic Azad University
I R Iran
mahdie.azari@gmail.com


F.
FalahatiNezhad
Safadasht Branch, Islamic Azad University
Safadasht Branch, Islamic Azad University
I R Iran
farzanehfalahati_n@yahoo.com
vertex degree
graph invariant
Splice
[[1] M. V. Diudea, QSPR/QSAR Studies by Molecular Descriptors, New York, NOVA,##[2] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer,##Berlin, 1986.##[3] N. Trinajstić, Chemical Graph Theory, CRC Press, Boca Raton, FL, 1992.##[4] M. Randić, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975)##6609–6615.##[5] B. Zhou and N. Trinajstić, On a novel connectivity index, J. Math. Chem. 46 (2009)##1252–1270.##[6] S. Fajtlowicz, On conjectures on Graffiti–II, Congr. Numer. 60 (1987) 187–197.##[7] E. Estrada, L. Torres, L. Rodriguez and I. Gutman, An atom–bond connectivity index:##Modelling the enthalpy of formation of alkanes, Indian J. Chem. 37A (1998) 849–855.##[8] E. Estrada, Atombond connectivity and the energetic of branched alkanes, Chem.##Phys. Lett. 463 (2008) 422–425.##[9] B. Furtula, A. Graovac and D. Vukičević, Augmented Zagreb index, J. Math. Chem. 48##(2) (2010) 370–380.##[10] D. Vukičević and B. Furtula, Topological index based on the ratios of geometrical and##arithmetical means of end–vertex degrees of edges, J. Math. Chem. 46 (2009) 1369–1376.##[11] H. Deng, J. Yang and F. Xia, A general modeling of some vertex–degree based##topological indices in benzenoid systems and phenylenes, Comput. Math. Appl. 61 (2011)##3017–3023.##[12] A. R. Ashrafi, A. Hamzeh and S. Hossein–Zadeh, Calculation of some topological##indices of splices and links of graphs, J. Appl. Math. Inf. 29 (1–2) (2011) 327–335.##[13] M. Azari, Sharp lower bounds on the NarumiKatayama index of graph operations,##Appl. Math. Comput. 239 C (2014) 409–421.##[14] M. Azari and A. Iranmanesh, Chemical graphs constructed from rooted product and##their Zagreb indices, MATCH Commun. Math. Comput. Chem. 70 (2013) 901–919. ##[15] M. Azari and A. Iranmanesh, Computing the eccentricdistance sum for graph##operations, Discrete Appl. Math. 161 (18) (2013) 2827–2840.##[16] M. Azari and A. Iranmanesh, Computing Wiener–like topological invariants for some##composite graphs and some nanotubes and nanotori, In: I. Gutman, (Ed.), Topics in##Chemical Graph Theory, Univ. Kragujevac, Kragujevac, 2014, pp. 69–90.##[17] M. Azari, A. Iranmanesh and I. Gutman, Zagreb indices of bridge and chain graphs,##MATCH Commun. Math. Comput. Chem. 70 (2013) 921–938.##[18] A. Iranmanesh, M. A. Hosseinzadeh and I. Gutman, On multiplicative Zagreb indices##of graphs, Iranian J. Math. Chem. 3(2) (2012) 145–154.##[19] M. Mogharrab and I. Gutman, Bridge graphs and their topological indices, MATCH##Commun. Math. Comput. Chem. 69 (2013) 579–587.##[20] R. Sharafdini and I. Gutman, Splice graphs and their topological indices, Kragujevac##J. Sci. 35 (2013) 89–98.##[21] T. Došlić, Splices, links and their degree–weighted Wiener polynomials, Graph##Theory Notes New York 48 (2005) 47–55.##[22] M. Azari, A note on vertex–edge Wiener indices, Iranian J. Math. Chem. 7(1) (2016)##11–17. ##]
An Upper Bound on the First Zagreb Index in Trees
2
2
In this paper we give sharp upper bounds on the Zagreb indices and characterize all trees achieving equality in these bounds. Also, we give lower bound on first Zagreb coindex of trees.
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71
82


R.
Rasi
Azarbaijan Shahid Madani University, Tabriz, Iran
Azarbaijan Shahid Madani University, Tabriz,
I R Iran


S.
Sheikholeslami
Azarbaijan Shahid Madani University, Tabriz, Iran
Azarbaijan Shahid Madani University, Tabriz,
I R Iran


A.
Behmaram
Institute for Research in Fundamental Sciences, Tehran, Iran
Institute for Research in Fundamental Sciences,
I R Iran
behmarammath@gmail.com
First Zagreb index
First Zagreb coindex
tree
Chemical tree
[[1] A. R. Ashrafi, T. Došlić, A. Hamzeh, The Zagreb coindices of graph operations,##Discrete Appl. Math. 158 (2010), 1571–1578.##[2] K. C. Das, Sharp bounds for the sum of the squares of the degrees of a graph,##Kragujevac J. Math. 25 (2003), 31–49.##[3] K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math.##285 (2004), 57–66.##[4] D. de Caen, An upper bound on the sum of squares in a graph, Discrete Math. 185##(1998), 245–248.##[5] T. Došlić, Vertexweighted Wiener polynomials for composite graphs, Ars. Math.##Contemp. 1 (2008), 66–80.##[6] I. Gutman, N. Trinajstić, Graph theory and molecular orbitals. Total electron energy##of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538.##[7] X. L. Li, I. Gutman, Mathematical Aspects of RandićType Molecular Structure##Descriptors, Mathematical Chemistry Monograph 1, University of Kragujevac, 2006.##[8] S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, The Zagreb indices 30 years after,##Croat. Chem. Acta 76 (2003), 113–124.##[9] Ž. Kovijanić Vukićević, G. Popivoda, Chemical trees with extreme values of Zagreb##indices and coindices, Iranian. J. Math. Chem. 5 (2014) 19–29.##[10] S. Zhang, W. Wang, T. C. E. Cheng, Bicyclic graphs with the first three smallest and##largest values of the first general Zagreb index, MATCH Commun. Math. Comput. Chem.##56 (2006), 579–592.##[11] B. Zhou, I. Gutman, Relations between Wiener, hyperWiener and Zagreb indices,##Chem. Phys. Lett. 394 (2004), 93–95.##]
DistanceBased Topological Indices and Double graph
2
2
Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closedform formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topological indices.
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83
91


M.
Jamil
ABDUS SALAM SCHOOL OF MATHEMATICAL SCIENCES, GOVERNMENT COLLEGE UNIVERSITY, LAHORE, PAKISTAN.
ABDUS SALAM SCHOOL OF MATHEMATICAL SCIENCES,
Pakistan
m.kamran.sms@gmail.com
Wiener index
Harary index
Double graph
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