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Characterisation of forests with trivial game domination numbers
Nadjafi-Arani, M. J.,Siggers, M.,Soltani, H. Springer Science + Business Media 2016 Journal of combinatorial optimization Vol.32 No.3
<P>In the domination game, two players, the Dominator and Staller, take turns adding vertices of a fixed graph to a set, at each turn increasing the number of vertices dominated by the set, until the final set dominates the whole graph. The Dominator plays to minimise the size of the set while the Staller plays to maximise it. A graph is -trivial if when the Dominator plays first and both players play optimally, the set is a minimum dominating set of the graph. A graph is -trivial if the same is true when the Staller plays first. We consider the problem of characterising -trivial and -trivial graphs. We give complete characterisations of -trivial forests and of -trivial forests. We also show that -connected -trivial graphs cannot have large girth, and conjecture that the same holds without the connectivity condition.</P>
On maximum Wiener index of trees and graphs with given radius
Das, K. C.,Nadjafi-Arani, M. J. Springer Science + Business Media 2017 Journal of combinatorial optimization Vol.34 No.2
<P>Let G be a connected graph of order n. The long-standing open and close problems in distance graph theory are: what is the Wiener index W(G) or average distance mu(G) among all graphs of order n with diameter d (radius r)? There are very few number of articles where were worked on the relationship between radius or diameter and Wiener index. In this paper, we give an upper bound on Wiener index of trees and graphs in terms of number of vertices n, radius r, and characterize the extremal graphs. Moreover, from this result we give an upper bound on mu(G) in terms of order and independence number of graph G. Also we present another upper bound on Wiener index of graphs in terms of number of vertices n, radius r and maximum degree Delta, and characterize the extremal graphs.</P>
Relations between distance-based and degree-based topological indices
Das, K.Ch.,Gutman, I.,Nadjafi-Arani, M.J. Elsevier [etc.] 2015 Applied Mathematics and Computation Vol.270 No.-
Let W, Sz, PI, and WP be, respectively, the Wiener, Szeged, PI, and Wiener polarity indices of a molecular graph G. Let M<SUB>1</SUB> and M<SUB>2</SUB> be the first and second Zagreb indices of G. We obtain relations between these classical distance- and degree-based topological indices.