Steiner Degree Distance of Two Graph Products

Mao, Yaping; Wang, Zhao; Das, Kinkar Ch.

doi:10.2478/auom-2019-0020

Abstract

The degree distance DD(G) of a connected graph G was invented by Dobrynin and Kochetova in 1994. Recently, one of the present authors introduced the concept of k-center Steiner degree distance defined as ${SDD}_{k} (G) = \sum_{\begin{matrix} S \subseteq V (G) \\ | S | = k \end{matrix}} [\sum_{v \in S} \deg_{G} (v)] d_{G} (S),$ SDD_k (G) = \sum\limits_{\mathop {S \subseteq V(G)}\limits_{\left| S \right| = k} } {\left[ {\sum\limits_{v \in S} {{\it deg} _G (v)} } \right]d_G (S),} where d_G(S) is the Steiner k-distance of S and deg_G(v) is the degree of the vertex v in G. In this paper, we investigate the Steiner degree distance of complete and Cartesian product graphs.

References

[1] P. Ali, P. Dankelmann, S. Mukwembi, Upper bounds on the Steiner diameter of a graph, Discr. Appl. Math.160 (2012) 1845–1850.10.1016/j.dam.2012.03.031
Search in Google Scholar Back to article
[2] P. Ali, S. Mukwembi, S. Munyira, Degree distance and vertex–connectivity, Discr. Appl. Math.161 (2013) 2802–2811.10.1016/j.dam.2013.06.033
Search in Google Scholar Back to article
[3] P. Ali, S. Mukwembi, S. Munyira, Degree distance and edge–connectivity, Australas. J. Comb.60 (2014) 50–68.
Search in Google Scholar Back to article
[4] M. An, L. Xiong, K.C. Das, Two upper bounds for the degree distances of four sums of graphs, Filomat28 (2014) 579–590.10.2298/FIL1403579A
Search in Google Scholar Back to article
[5] J. A. Bondy, U. S. R. Murty, Graph Theory, Springer, New York, 2008.10.1007/978-1-84628-970-5
Search in Google Scholar Back to article
[6] O. Bucicovschi, S. M. Cioabă, The minimum degree distance of graphs of given order and size, Discr. Appl. Math.156 (2008) 3518–3521.10.1016/j.dam.2008.03.036
Search in Google Scholar Back to article
[7] F. Buckley, F. Harary, Distance in Graphs, Addison–Wesley, Redwood, 1990.
Search in Google Scholar Back to article
[8] J. Cáceresa, A. Márquezb, M. L. Puertasa, Steiner distance and convexity in graphs, Eur. J. Comb.29 (2008) 726–736.10.1016/j.ejc.2007.03.007
Search in Google Scholar Back to article
[9] G. Chartrand, O. R. Oellermann, S. Tian, H. B. Zou, Steiner distance in graphs, Časopis Pest. Mat.114 (1989) 399–410.10.21136/CPM.1989.118395
Search in Google Scholar Back to article
[10] P. Dankelmann, O. R. Oellermann, H. C. Swart, The average Steiner distance of a graph, J. Graph Theory22 (1996) 15–22.10.1002/(SICI)1097-0118(199605)22:1<;15::AID-JGT3>3.0.CO;2-O
Search in Google Scholar Back to article
[11] A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and application, Acta Appl. Math.66 (2001) 211–249.10.1023/A:1010767517079
Search in Google Scholar Back to article
[12] A. A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Wiener index of hexagonal systems, Acta Appl. Math.72 (2002) 247–294.10.1023/A:1016290123303
Search in Google Scholar Back to article
[13] A. Dobrynin, A. Kochetova, Degree distance of a graph: A degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci.34 (1994) 1082–1086.10.1021/ci00021a008
Search in Google Scholar Back to article
[14] W. Goddard, O. R. Oellermann, Distance in graphs, in: M. Dehmer (Ed.), Structural Analysis of Complex Networks, Birkhäuser, Dordrecht, 2011, pp. 49–72.10.1007/978-0-8176-4789-6_3
Search in Google Scholar Back to article
[15] T. Gologranc, Steiner Convex Sets and Cartesian Product, Bull. Malays. Math. Sci. Soc. DOI 10.1007/s40840-016-0312-8.10.1007/s40840-016-0312-8
Open DOI Search in Google Scholar Back to article
[16] I. Gutman, On Steiner degree distance of trees, Appl. Math. Comput.283 (2016) 163–167.10.1016/j.amc.2016.02.038
Search in Google Scholar Back to article
[17] I. Gutman, On two degree-and-distance-based graph invariants, Bull. Acad. Serbe Sci. Arts (Cl. Sci. Math. Natur.), in press.
Search in Google Scholar Back to article
[18] I. Gutman, B. Furtula, K. C. Das, On some degree–and–distance–based graph invariants of trees, Appl. Math. Comput.289 (2016) 1–6.10.1016/j.amc.2016.04.040
Search in Google Scholar Back to article
[19] I. Gutman, Y. N. Yeh, S. L. Lee, Y. L. Luo, Some recent results in the theory of the Wiener number, Indian J. Chem.32A (1993) 651–661.
Search in Google Scholar Back to article
[20] M. Knor, R. Škrekovski, A. Tepeh, Mathematical aspects of Wiener index, Ars Math. Contemp.11 (2016) 327–352.10.26493/1855-3974.795.ebf
Search in Google Scholar Back to article
[21] X. Li, Y. Mao, I. Gutman, The Steiner Wiener index of a graph, Discuss. Math. Graph Theory36 (2016) 455–465.10.7151/dmgt.1868
Search in Google Scholar Back to article
[22] X. Li, Y. Mao, I. Gutman, Inverse problem on the Steiner Wiener index, Discuss. Math. Graph Theory38 (2018) 83–95.10.7151/dmgt.2000
Search in Google Scholar Back to article
[23] Y. Mao, The Steiner diameter of a graph, Bull. Iran. Math. Soc.43 (2) (2017) 439–454.
Search in Google Scholar Back to article
[24] Y. Mao, E. Cheng, Z. Wang, Steiner distance in product networks, arXiv:1703.01410 [math.CO] 2017.
Search in Google Scholar Back to article
[25] Y. Mao, K. C. Das, Steiner Gutman index, MATCH Commun. Math. Comput. Chem.79 (3) (2018) 779–794.
Search in Google Scholar Back to article
[26] Y. Mao, Z. Wang, I. Gutman, Steiner Wiener index of graph products, Trans. Combin.5 (3) (2016) 39–50.
Search in Google Scholar Back to article
[27] Y. Mao, Z. Wang, I. Gutman, A. Klobučar, Steiner degree distance, MATCH Commun Math. Comput. Chem.78 (1) (2017) 221–230.
Search in Google Scholar Back to article
[28] Y. Mao, Z. Wang, I. Gutman, H. Li, Nordhaus-Gaddum-type results for the Steiner Wiener index of graphs, Discrete Appl. Math.219 (2017) 167–175.10.1016/j.dam.2016.11.014
Search in Google Scholar Back to article
[29] Y. Mao, Z. Wang, Y. Xiao, C. Ye, Steiner Wiener index and connectivity of graphs, Utilitas Math.102 (2017) 51–57.
Search in Google Scholar Back to article
[30] S. Mukwembi, S. Munyira, Degree distance and minimum degree, Bull. Austral. Math. Soc.87 (2013) 255–271.10.1017/S0004972712000354
Search in Google Scholar Back to article
[31] O. R. Oellermann, S. Tian, Steiner centers in graphs, J. Graph Theory14 (1990) 585–597.10.1002/jgt.3190140510
Search in Google Scholar Back to article
[32] K. Pattabiraman, P. Kandan, Generalization of the degree distance of the tensor product of graphs, Australas. J. Comb.62 (2015) 211–227.
Search in Google Scholar Back to article
[33] D. H. Rouvray, Harry in the limelight: The life and times of Harry Wiener, in: D. H. Rouvray, R. B. King (Eds.), Topology in Chemistry – Discrete Mathematics of Molecules, Horwood, Chichester, 2002, pp. 1–15.10.1016/B978-1-898563-76-1.50005-6
Search in Google Scholar Back to article
[34] D. H. Rouvray, The rich legacy of half century of the Wiener index, in: D. H. Rouvray, R. B. King (Eds.), Topology in Chemistry – Discrete Mathematics of Molecules, Horwood, Chichester, 2002, pp. 16–37.10.1016/B978-1-898563-76-1.50006-8
Search in Google Scholar Back to article
[35] V. Sheeba Agnes, Degree distance and Gutman index of corona product of graphs, Trans. Comb.4 (3) (2015) 11–23.
Search in Google Scholar Back to article
[36] K. Xu, M. Liu, K. C. Das, I. Gutman, B. Furtula, A survey on graphs extremal with respect to distance–based topological indices, MATCH Commun. Math. Comput. Chem.71 (2014) 461–508.
Search in Google Scholar Back to article

Steiner Degree Distance of Two Graph Products

Abstract

Paradigm

My account