The topological indices are functions on the graph that do not depend on the labeling of their vertices. They are used by chemists for studying the properties of chemical compounds.  Let  be a simple connected graph. The Hyper-Zagreb index of the graph ,  is defined as  ,where  and  are the degrees of vertex  and , respectively. In this paper, we study the Hyper-Zagreb index and give upper and lower bounds for .

This study considers a k-objective time-varying shortest path problem, which cannot be combined into a single overall objective. In this problem, the transit cost to traverse an arc is varying over time, which depend upon the departure time at the beginning vertex of the arc. An algorithm is presented for finding the efficient solutions of problem and its complexity of algorithm is analyzed. Finally, an illustrative example is also provided to clarify the problem.Key words: Time-varying optimization, K-criteria shortest path problem INTRODUCTIONSpecial form of the bi-criteria path problems was introduced by Hansen (1980). The number of pareto paths set were defined, where it grew exponentially with the number of nodes set. He solved the monotone bi-criteria and the bi-objective path problems by using a label setting algorithm. A multiple label setting algorithm was expanded to generate pareto shortest path by Martins (1984). Moreover, Corley and Moon (1985) applied the dynamic programming to solve the multi-criteria shortest path problem. Brumbaugh-Smith and Shier (1989) proposed the linear time algorithm to solve the bi-criteria shortest path problems. Several algorithm to solve bi-objective multi-model shortest paths by using bidirectional search were presented by Artigues et al. (2013), where path viability constraints are modeled by a finite state automaton. A comprehensive survey on multi-criteria shortest path algorithm is given by Ehrgott and Gandibleux (2002). The reader is referred to Ahuja et al. (1993), Artigues et al. (2013), Bertsekas (1991, Getachew et al. (2000), Reinhardt andPisinger (2011) andSchrijver (2003) for developments in that area.In time-dependent version of problem, Cooke and Halsey (1966) described the fastest path problem. Kostreva and Wiecek (1993) extended the research of Cooke and Halsey (1966) to the multi-criteria case. Getachew et al. (2000) developed the results of Kostreva and Wiecek (1993). Moreover, they replaced the non-decreasing arc costs constraints by bounds on the cost and relaxed the time grid constraints. Two-objective function with time dependent data was studied by Hamacher et al. (2006). The problem called the time-dependent bi-criteria shortest path problem. Moreover, they reviewed an algorithm proposed by Kostreva and Wiecek (1993), presented a new label setting algorithm and compared both algorithms numerically. Sha and Wong (2007) considered the best path with multi-criteria in time-varying network, where a transit time b (x, y, t) is needed to traverse an arc (x, y). They supposed the time-varying version of the minimum cost-reliability ratio path problem.

Abstract. In this paper, we focus on the time-varying shortest path problem, where the transit costs are fuzzy numbers. Moreover, we consider this problem in which the transit time can be shortened at a fuzzy speedup cost. Speedup may also be a better decision to find the shortest path from a source vertex to a specified vertex.

In this paper, by providing some different conditions respect to another works, we shall present two results on absolute retractivity of some sets related to some multifunctions of the form F : X × X → P b,cl (X), on complete metric spaces.

In this paper, the maximum capacity path problem in time-varying network is presented, where waiting times at vertices are not allowable. Moreover, the capacities are considered the generalized trapezoidal fuzzy number. An exact algorithm is proposed which can find a optimal solution of problem subject to the time of path is at most T, where T is a given time horizon.