ORIGINAL_ARTICLE
The central vertices and radius of the regular graph of ideals
The regular graph of ideals of the commutative ring $R$, denoted by ${\Gamma_{reg}}(R)$, is a graph whose vertex set is the set of all non-trivial ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if either $I$ contains a $J$-regular element or $J$ contains an $I$-regular element. In this paper, it is proved that the radius of $\Gamma_{reg}(R)$ equals $3$. The central vertices of $\Gamma_{reg}(R)$ are determined, too.
http://toc.ui.ac.ir/article_21472_57a7aea214c4516a524744b78f00943a.pdf
2017-12-01T11:23:20
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1
13
10.22108/toc.2017.21472
Arc
artinian ring
eccentricity
radius
regular digraph
Farzad
Shaveisi
f.shaveisi@ipm.ir
true
1
Razi University
Razi University
Razi University
LEAD_AUTHOR
[1] G. Aalip our, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shaveisi, Minimal prime ideals and cycles in annihilating-ideal graphs, Rocky Mountain J. Math., 43 no. 5 (2013) 1415-1425.
1
[2] M. Afkhami, M. Karimi and K. Khashayarmanesh, On the regular digraph of ideals of a commutative ring, Bul l. Aust. Math. Soc., 88 no. 2 (2012) 177-189.
2
[3] M. F. Atiyah, I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Company, 1969.
3
[4] J. A. Bondy, U. S. R. Murty, Graph theory with applications , American Elsevier, New York, 1976.
4
[5] I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309 (2009) 5381-5392.
5
[6] H. R. Dorbidi and R. Manaviyat, Some results on the comaximal ideal graph of a commutative ring, Trans. Comb., 5 no. 4 (2016) 9-20.
6
[7] M. J. Nikmehr and F. Shaveisi, The regular digraph of ideals of a commutative ring, Acta Math. Hungar., 134 (2012) 516-528.
7
[8] F. Shaveisi, Some results on the annihilating-ideal graphs, Canad. Math. Bul l., 59 no. 3 (2016) 641-651.
8
[9] F. Shaveisi and R. Nikandish, The nil-graph of ideals of a commutative ring, Bul l. Malays. Math. Sci. Soc. (2), 39 no. 1 (2016) 3{11.
9
[10] T. Tamizh Chelvman and S. Nithya, A note on the zero divisor graph of a latice, Trans. Comb., 3 no. 3 (2014) 51-59.
10
ORIGINAL_ARTICLE
The harmonic index of subdivision graphs
The harmonic index of a graph $G$ is defined as the sum of the weights $\frac{2}{\deg_G(u)+\deg_G(v)}$ of all edges $uv$ of $G$, where $\deg_G(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we study the harmonic index of subdivision graphs, $t$-subdivision graphs and also, $S$-sum and $S_t$-sum of graphs.
http://toc.ui.ac.ir/article_21471_6d4574ac2fe03052a0872fb991c96309.pdf
2017-12-01T11:23:20
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15
27
10.22108/toc.2017.21471
harmonic index
subdivision
$S$-sum
inverse degree
Zagreb index
Bibi Naimeh
Onagh
bn.onagh@gu.ac.ir
true
1
Golestan University
Golestan University
Golestan University
LEAD_AUTHOR
[1] A. Astaneh-Asl and G. H. Fath-Tabar, Computing the rst and third Zagreb p olynomials of Cartesian pro duct of graphs, Iranian J. Math. Chem., 2 no. 2 (2011) 73-78.
1
[2] P. S. Bullen, A dictionary of inequalities, Addison-Wesley Longman, 1998.
2
[3] H. Deng, S. Balachandran, S. K. Ayyaswamy and Y. B. Venkatakrishnan, On the harmonic index and the chromatic numb er of a graph, Discrete Appl. Math., 161 (2013) 2740-2744.
3
[4] M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math., 157 (2009) 794-803.
4
[5] S. Fa jtlowicz, On conjectures of graﬃti I I, Congr. Numer., 60 (1987) 189-197.
5
[6] G. H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem., 65 (2011) 79-84.
6
[7] A. Ilic, M. Ilic and B. Liu, On the upp er b ounds for the rst Zagreb index, Kragujevac J. Math., 35 (2011) 173-182.
7
[8] J. Li, J. B. Lv and Y. Liu, The harmonic index of some graphs, Bul l. Malays. Math. Sci. Soc., 39 (2016) 331-340.
8
[9] J. B. Lv and J. Li, On the harmonic index and the matching numb ers of trees, Ars Combin., 116 (2014) 407-416.
9
[10] J. B. Lv, J. Li and W. C. Shiu, The harmonic index of unicyclic graphs with given matching number, Kragujevac J. Math., 38 (2014) 173-182.
10
[11] S. Nikolic, G. Kovacevic, A. Milicevic and N. Trina jstic, The Zagreb indices 30 years after, Croat. Chem. Acta, 76 (2003) 113-124.
11
[12] B. N. Onagh, The harmonic index of pro duct graphs, Math. Sci., (2017) 1-7.
12
[13] B. N. Onagh, The harmonic index of edge-semitotal graphs, total graphs and related sums, Kragujevac J. Math., to app ear.
13
[14] B. N. Onagh, The harmonic index for R-sum of graphs, submitted.
14
[15] B. S. Shwetha, V. Lokesha and P. S. Ranjini, On the harmonic index of graph op erations, Trans. Comb. , 4 no. 4 (2015) 5-14.
15
[16] R. Wu, A. Tang and H. Deng, A lower b ound for the harmonic index of a graph with minimum degree at least two, Filomat, 27 (2013) 51-55.
16
[17] X. Xu, Relationships b etween harmonic index and other top ological indices, Appl. Math. Sci., 6 (2012) 2013-2018.
17
[18] L. Zhong, The harmonic index for graphs, Appl. Math. Lett., 25 (2012) 561-566.
18
[19] L. Zhong, The harmonic index on unicyclic graphs, Ars Combin., 104 (2012) 261-269.
19
[20] L. Zhong, On the harmonic index and the girth for graphs, Rom. J. Inf. Sci. Tech., 16 no. 4 (2013) 253-260.
20
[21] L. Zhong and K. Xu, The harmonic index for bicyclic graphs, Utilitas Math., 90 (2013) 23-32.
21
[22] L. Zhong and K. Xu, Inequalities b etween vertex-degree-based top ological indices, MATCH Commun. Math. Com-put. Chem., 71 (2014) 627-642.
22
ORIGINAL_ARTICLE
Splices, Links, and their Edge-Degree Distances
The edge-degree distance of a simple connected graph G is defined as the sum of the terms (d(e|G)+d(f|G))d(e,f|G) over all unordered pairs {e,f} of edges of G, where d(e|G) and d(e,f|G) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link.
http://toc.ui.ac.ir/article_21614_033f4714ff9a47c358a450a46e9a3122.pdf
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29
42
10.22108/toc.2017.21614
Distance
degree
edge-degree distance
splice of graphs
link of graphs
Mahdieh
Azari
mahdie.azari@gmail.com
true
1
Kazerun Branch, Islamic Azad University
Kazerun Branch, Islamic Azad University
Kazerun Branch, Islamic Azad University
LEAD_AUTHOR
Hojjatollah
Divanpour
h.divanpour@yahoo.com
true
2
Shiraz Technical College, Technical and Vocational University
Shiraz Technical College, Technical and Vocational University
Shiraz Technical College, Technical and Vocational University
AUTHOR
[1] P. Ali and S. Mukwembi, Degree distance and edge-connectivity, Australas. J. Combin., 60 no. 1 (2014) 50-68.
1
[2] A. R. Ashra, A. Hamzeh and S. Hossein-Zadeh, Calculation of some top ological indices of splices and links of graphs, J. Appl. Math. Inf., 29 no. 1-2 (2011) 327-335.
2
[3] M. Azari, Sharp lower b ounds on the Narumi-Katayama index of graph op erations, Appl. Math. Comput., 239 (2014) 409-421.
3
[4] M. Azari, A note on vertex-edge Wiener indices, Iranian J. Math. Chem., 7 no. 1 (2016) 11-17.
4
[5] M. Azari and F. Falahati-Nezhad, On vertex-degree-based invariants of link of graphs with application in dendrimers, J. Comput. Theor. Nanosci., 12 no. 12 (2015) 5611-5616.
5
[6] M. Azari and F. Falahati-Nezhad, Splice graphs and their vertex-degree-based invariants, Iranian J. Math. Chem., 8 no. 1 (2017) 61-70.
6
[7] M. Azari, A. Iranmanesh and I. Gutman, Zagreb indices of bridge and chain graphs, MATCH Commun. Math. Comput. Chem., 70 no. 3 (2013) 921-938.
7
[8] O. Bucicovschi and S. M. Cioaba, The minimum degree distance of graphs of given order and size, Discrete Appl. Math., 156 no. 18 (2008) 3518-3521.
8
[9] P. Dankelmann, I. Gutman, S. Mukwembi and H. C. Swart, On the degree distance of a graph, Discrete Appl. Math., 157 no. 13 (2009) 2773-2777.
9
[10] K. C. Das, G. Su and L. Xiong, Relation b etween degree distance and Gutman index of graphs, MATCH Commun. Math. Comput. Chem., 76 no. 1 (2016) 221-232.
10
[11] M. V. Diudea, QSPR/QSAR Studies by Molecular Descriptors, NOVA, New York, 2001.
11
[12] A. Dobrynin and A. A. Ko chetova, Degree distance of a graph: A degree a nalogue of the Wiener index, J. Chem. Inf. Comput. Sci., 34 no. 5 (1994) 1082-1086.
12
[13] T. Doslic, Splices, links and their degree-weighted Wiener p olynomials, Graph Theory Notes N. Y., 48 (2005) 47-55.
13
[14] Z. Du and B. Zhou, Degree distance of unicyclic graphs, Filomat, 24 no. 4 (2010) 95-120.
14
[15] I. Gutman, Selected prop erties of the Schultz molecular top ological index, J. Chem. Inf. Comput. Sci., 34 no. 5 (1994) 1087-1089.
15
[16] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, 1986.
16
[17] I. Gutman and N. Trina jstic, Graph theory and molecular orbitals, Total-electron energy of alternant hydro car-b ons, Chem. Phys. Lett., 17 no. 4 (1972) 535-538.
17
[18] A. Hamzeh, A. Iranmanesh, M. A. Hosseinzadeh and S. Hossein-Zadeh, The Hosoya index and the Merrield-Simmons index of some graphs, Trans. Comb., 1 no. 4 (2012) 51-60.
18
[19] M. A. Hosseinzadeh, A. Iranmanesh and T. Doslic, On the Narumi-Katayama index of splice and link of graphs, Electron. Notes Discrete Math., 45 (2014) 141-146.
19
[20] Y. Hou and A. Chang, The unicyclic graphs with maximum degree distance, J. Math. Study, 39 (2006) 18-24.
20
[21] A. Ilic, S. Klavzar and D. Stevanovic, Calculating the degree distance of partial Hamming graphs, MATCH Commun.Math. Comput. Chem., 63 no. 2 (2010) 411-424.
21
[22] A. Ilic, D. Stevanovic, L. Feng, G. Yu and P. Dankelmann, Degree distance of unicyclic and bicyclic graphs, Discrete Appl. Math., 159 no. 8 (2011) 779-788.
22
[23] A. Iranmanesh, I. Gutman, O. Khormali and A. Mahmiani, The edge versions of the Wiener index, MATCH Commun. Math. Comput. Chem., 61 no. 3 (2009) 663-672.
23
[24] A. Iranmanesh, O. Khormali and A. Ahmadi, Generalized edge-Schultz indices of some graphs, MATCH Commun. Math. Comput. Chem., 65 no. 1 (2011) 93-112.
24
[25] M. Mogharrab and I. Gutman, Bridge graphs and their top ological indices, MATCH Commun. Math. Comput. Chem., 69 no. 3 (2013) 579-587.
25
[26] R. Sharafdini and I. Gutman, Splice graphs and their top ological indices, Kragujevac J. Sci., 35 (2013) 89-98.
26
[27] I. Tomescu, Prop erties of connected graphs having minimum degree distance, Discrete Math., 309 no. 9 (2009) 2745-2748.
27
[28] I. Tomescu, Ordering connected graphs having small degree distances, Discrete Appl. Math., 158 no. 15 (2010) 1714-1717.
28
[29] I. Tomescu and S. Kanwal, Ordering connected graphs having small degree distances II, MATCH Commun. Math. Comput. Chem., 67 no. 2 (2012) 425-437.
29
[30] H. Wiener, Structural determination of paraﬃn b oiling p oints, J. Am. Chem. Soc., 69 no. 1 (1947) 17-20.
30
ORIGINAL_ARTICLE
On the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue of a graph
The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc\left(G\right)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The harmonic index $H\left(G\right)$ of a graph $G$ is defined as the sum of $\frac{2}{d_{i}+d_{j}}$ over all edges $v_{i}v_{j}$ of $G$, where $d_{i}$ denotes the degree of a vertex $v_{i}$ in $G$. In this paper, we determine the unique tree with minimum average eccentricity among the set of trees with given number of pendent vertices and determine the unique tree with maximum average eccentricity among the set of $n$-vertex trees with two adjacent vertices of maximum degree $\Delta$, where $n\geq 2\Delta$. Also, we give some relations between the average eccentricity, the harmonic index and the largest signless Laplacian eigenvalue, and strengthen a result on the Randi\'{c} index and the largest signless Laplacian eigenvalue conjectured by Hansen and Lucas \cite{hl}.
http://toc.ui.ac.ir/article_21470_6107bccf810358fdefb9471c7d0ba0a8.pdf
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50
10.22108/toc.2017.21470
Average eccentricity
harmonic index
signless Laplacian eigenvalue
extremal value
Hanyuan
Deng
hydeng@hunnu.edu.cn
true
1
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
College of Mathematics and Computer Science, Hunan Normal University, Changsha, Hunan 410081, P. R. China
LEAD_AUTHOR
S.
Balachandran
bala@maths.sastra.edu
true
2
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
AUTHOR
S. K.
Ayyaswamy
true
3
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
AUTHOR
Y. B.
Venkatakrishnan
venkatakrish2@maths.sastra.edu
true
4
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
Department of Mathematics, School of Humanities and Sciences,SASTRA University, Thanjavur, India
AUTHOR
[1] Y. Chen, Prop erties of sp ectra of graphs and line graphs, Appl. Math. J. Ser. B, 3 (2002) 371-376.
1
[2] D. Cvetkovic, P. Rowlinson and S. K. Simic, Eigenvalue b ounds for the signless Laplacian, Publ. Inst. Math. (Beograd) (N. S), 81 (95) (2007) 11-27.
2
[3] P. Dankelmann, W. Go ddard and C. S. Swart, The average eccentricity of a graph and its subgraph, Util. Math., 65 (2004) 41-51.
3
[4] H. Deng, S. Balachandran, S. K. Ayyaswamy and Y. B. Venkatakrishnan, On the harmonic index and the chromatic numb er of a graph, Discrete Appl. Math., 161 (2013) 2740-2744.
4
[5] H. Deng, S. Balachandran, S. K. Ayyaswamy and Y. B. Venkatakrishnan, On harmonic indices of trees, unicyclic graphs and bicyclic graphs, ARS Combinatoria , CXXX (2017) 239-248.
5
[6] A. A. Dobrynin, R. C. Entringer and I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math., 66 (2001) 211-249.
6
[7] Z. Du and A. Ilic, On AGX conjectures regarding average eccentricity, MATCH Commun. Math. Comput. Chem., 69 (2013) 597-609.
7
[8] S. Fa jtlowicz, On conjectures of Graﬃti-I I, Cong. Numer., 60 (1987) 187{197.
8
[9] O. Favaron, M. Mahio and J. F. Sacle, Some eigenvalue prop erties in graphs (Conjectures of Graﬃti-II), Discrete Math., 111 (1993) 197-220.
9
[10] L. Feng and G. Yu, On three conjectures involving the signless Laplacian sp ectral radius of graphs, Publ. Inst. Math. (Beograd) (N.S), 85 (99) (2009) 35-38.
10
[11] P. Hansen, C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl., 432 (2010) 3319-3336.
11
[12] P. Hansen, D. Vukicevic, Variable neighb orho o d search for extremal graphs. 23. On the Randic index and the chromatic numb er, Discrete Math., 309 (2009) 4228-4234.
12
[13] A. Ilic, Note on the harmonic index of a graph, Arxiv preprint arXiv: 1204.3313, (2012).
13
[14] A. Ilic, Eccentric connectivity index, Novel Molecular Structure Descriptors-Theory and Applications II, Univ. Kragujevac, Kragujevac, 2010 139-168.
14
[15] A. Ilic and I. Gutman, Eccentric connectivity index of chemical trees, MATCH Commun. Math. Comput. Chem., 65 (2011) 731{744.
15
[16] M. K. Khalifeh, H. Youse-Azari, A. R. Ashra and S. G. Wagner, Some new results on distance-based graph invariants, European J. Combin., 30 (2009) 1149-1163.
16
[17] M. J. Morgan, S. Mukwembi and H. C. Swart, On the eccentric connectivity index of a graph, Discrete Math., 311 (2011) 1229-1234.
17
[18] V. Sharma, R. Goswami and A. K. Madan, Eccentric connectivity index: A noval highly discriminating top ological descriptor for structure-prop erty and structure-activity studies, J. Chem. Inf. Comput. Sci., 37 (1997) 273-282.
18
[19] Y. Tang and B. Zhou, On average eccentricity, MATCH Commun. Math. Comput. Chem., 67 (2012) 405-423.
19
[20] R. Wu, Z. Tang and H. Deng, A lower b ound for the harmonic index of a graph with minimum degree at least two, Filomat, 27 (1) (2013) 49-53.
20
[21] R. Wu, Z. Tang and H. Deng, On the harmonic index and the girth of a graph, Utilitas Math., 91 (2013) 65-69.
21
[22] L. Zhong, The harmonic index for graphs, Appl. Math. Lett., 25 (2012) 561-566.
22
[23] B. Zhou and Z. Du, On eccentric connectivity index, MATCH Commun. Math. Comput. Chem., 63 (2010) 181-198.
23
ORIGINAL_ARTICLE
Some topological indices and graph properties
In this paper, by using the degree sequences of graphs, we present sufficient conditions for a graph to be Hamiltonian, traceable, Hamilton-connected or $k$-connected in light of numerous topological indices such as the eccentric connectivity index, the eccentric distance sum, the connective eccentricity index.
http://toc.ui.ac.ir/article_21467_d05e2410fc3d5c5560f3430866b8af0e.pdf
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51
65
10.22108/toc.2017.21467
Topological indices
degree sequences
graph properties
Xiaomin
Zhu
962186133@qq.com
true
1
College of Science, Nantong University, Nantong, China
College of Science, Nantong University, Nantong, China
College of Science, Nantong University, Nantong, China
AUTHOR
Lihua
Feng
fenglh@163.com
true
2
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
AUTHOR
Minmin
Liu
903069441@qq.com
true
3
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
AUTHOR
Weijun
Liu
wjliu6210@126.com
true
4
School of Science, Nantong University, Nantong,226019， China,
School of Science, Nantong University, Nantong,226019， China,
School of Science, Nantong University, Nantong,226019， China,
LEAD_AUTHOR
Yuqin
Hu
1120233887@qq.com
true
5
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
School of Mathematics and Statistics, Central South University
New Campus, Changsha, Hunan, China.
AUTHOR
[1] D. Bauer, H. J. Bro ersma, J. van den Heuvel, N. Kahl, A. Nevo, E. Schmeichel, D. R. Wo o dall and M. Yatauro, Best monotone degree conditions for graph prop erties: a survey, Graphs Combin., 31 (2015) 1-22.
1
[2] C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973.
2
[3] J. A. Bondy, Prop erties of graphs with constraints on degrees, Studia Sci. Math. Hungar., 4 (1969) 473-475.
3
[4] W. Byer and D. Smeltzer, Edge b ounds in nonhamiltonian k -connected graphs, Discrete Math., 307 (2007) 1572-1579.
4
[5] L. Chen, J. Liu and Y. Shi, Matching energy of unicyclic and bicyclic graphs with a given diameter, Complexity, 21 (2015) 224-238.
5
[6] L. Chen and Y. T. Shi, The maximal matching energy of tricyclic graphs, MATCH Commun. Math. Comput. Chem., 73 (2015) 105-120.
6
[7] Z. Chen, M. Dehmer, Y. Shi and H. Yang, Sharp upp er b ounds for the Balaban index of bicyclic graphs, MATCH Commun. Math. Comput. Chem., 75 (2016) 105-128.
7
[8] V. Chvatal, On Hamiltons ideals, J. Combin. Theory Ser. B., 12 (1972) 163-168.
8
[9] A. Dobrynin, R. Entringerand I. Gutman, Wiener index of trees: theory and applications, Acta Appl. Math., 66 (2001) 211-249.
9
[10] S. Gupta, M. Singh and A. K. Madan, Connective eccentricity index: a novel top ological descriptor for predicting biological activity, J. Mol. Graph. Model., 18 (2000) 18{25.
10
[11] S. Gupta, M. Singh and A. K. Madan, Eccentric distance sum: A novel graph invariant for predicting biological and physical prop erties, J. Math. Anal. Appl., 275 (2002) 386{401.
11
[12] H. Hua and M. Wang, On Harary index and traceable graphs, MATCH Commun. Math. Comput. Chem., 70 (2013) 297-300.
12
[13] A. Ilic, G. H. Yu and L. H. Feng, On the eccentric distance sum of graphs, J. Math. Anal. Appl., 381 (2011) 590-600.
13
[14] O. Ivanciuc, QSAR comparative study of Wiener descriptors for weighted molecular graphs, J. Chem. Inf. Comput. Sci., 40 (2000) 1412-1422.
14
[15] R. Li, Harary index and some Hamiltonian prop erties of graphs, AKCE Inter. J. of Graphs and Comb., 12 (2015) 64-69.
15
[16] R. Li, Wiener index and some Hamiltonian prop erties of graphs, Inter. J. of Math. and Soft Computing, 5 (2015) 11-16.
16
[17] S. Li and Y. Song, On the sum of all distances in bipartite graphs, Discrete Appl. Math., 169 (2014) 176-185.
17
[18] S. C. Li, M. Zhang, G. H. Yu and L. H. Feng, On the extremal values of the eccentric distance sum of trees, J. Math. Anal. Appl., 390 (2012) 99-112.
18
[19] J. Ma, Y. Shi, Z. Wang and J. Yue, On Wiener p olarity index of bicyclic networks, Sci. Rep., 6 (2016) 19066. doi:10.1038/srep19066.
19
[20] J. Ma, Y.bT. Shi and Y. Yue, The Wiener p olarity index of graph pro ducts, Ars Combin., 116 (2014) 235{244.
20
[21] J. Plesnk, On the sum of all distances in a graph or diagraph, J. Graph Theory, 8 (1984) 1-21.
21
[22] V. Sharma, R. Goswami, A. K. Madan, Eccentric connectivity index: A novel highly discriminating top ological descriptor for structure prop erty and structure activity studies, J. Chem. Inf. Comput. Sci., 37 (1997) 273-282.
22
[23] S. Sardana, A. K. Madan, Predicting anti-HIV activity of TIBO derivatives: A computational approach using a novel top ological descriptor, J. Mol. Model., 8 (2002) 258-265.
23
[24] Y. Shi, Note on two generalizations of the Randic index, Appl. Math. Comput., 265 (2015) 1019-1025.
24
[25] L. Yang, Wiener index and traceable graphs, Bul l. Austral. Math. Soc., 88 (2013) 380-383.
25
[26] G.H. Yu, L.H. Feng, A. Ilic, On the eccentric distance sum of trees and unicyclic graphs, J. Math. Anal. Appl., 375 (2011) 99-07.
26
[27] G.H. Yu, H. Qu, L. Tang, L.H. Feng, On the connective eccentricity index of trees and unicyclic graphs with given diameter, J. Math. Aanl. Appl., 420 (2014) 1776-1786.
27
[28] B. Zhou, Z. Du, On eccentric connectivity index, MATCH Commun. Math. Comput. Chem., 63 (2010) 181-198.
28