The Wiener and detour indices of a molecular graph G are defined as the sum of the lengths of all shortest and longest paths between the vertices of G. In this paper the exact formulae for the Wiener and detour indices of a new type of nanostar dendrimers are given.

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Fuzzy sets and soft sets are two different soft computing models for representing vagueness and uncertainty. On the other domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This concept was introduced by [Benecke S, Cockayne EJ, Mynhardt CM. Secure total domination in graphs. Util Math. 2007;74:247-259.] in 2007 and Go and Canoy continue the study of these notions [Canoy RS, Go CE. Domination in the corona and join of graphs. Int Math Forum. 2011;6(16):763-771.] and afterward introduce total dominating and secure total dominating sets. Also graph operations like corona product play a very important role in mathematical chemistry, since some chemically interesting graphs can be obtained from some simpler graphs by different graph operations. In this paper, we characterised the dominating, total dominating, and secure total dominating sets in the corona of two fuzzy soft connected.

Sometimes information in a network model is based on multi-agent, multi-attribute, multi-object, multi-polar information or uncertainty rather than a single bit. An m-polar fuzzy model is useful for such network models which gives more and more precision, flexibility, and comparability to the system as compared to the classical, fuzzy models. On the other, The Steiner tree problem in networks, and particularly in graphs, was formulated by Hakimi [25] and Levi [21] by definition minimal size connected tree sub graph that contains the vertices in S. Steiner trees have applications to multiprocessor computer networks. For example, it may be desired to connect a certain set of processors with a sub network that uses the fewest communication links. In this paper, we extend Steiner distance SW k (G) for m-polar fuzzy graphs and give this parameter for Join, composition and Cartesian product of two m-polar fuzzy graphs.