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A hypertree is a special type of connected hypergraph that removes any, its hyperedge then results in a disconnected hypergraph. Relation between hypertrees (hypergraphs) and trees (graphs) can be helpful to solve real problems in hypernetworks and networks and it is the main tool in this regard. The purpose of this paper is to introduce a positive relation (as α-relation) on hypertrees that makes a connection between hypertrees and trees. This relation is dependent on some parameters such as path, length of a path, and the intersection of hyperedges. For any q ∈ N, we introduce the concepts of a derivable tree, (α, q)-hypergraph, and fundamental (α, q)-hypertree for the first time in this study and analyze the structures of derivable trees from hypertrees via given positive relation. In the final, we apply the notions of derivable trees, (α, q)-trees in real optimization problems by modeling hypernetworks and networks based on hypertrees and trees, respectively. [ABSTRACT FROM AUTHOR]

The deterministic permutation flow shop scheduling problem with makespan criterion is not solvable in polynomial time. Therefore, researchers have thought about heuristic algorithms. There are many heuristic algorithms for solving it that is a very important combinatorial optimization problem. In this paper, a new algorithm is proposed for solving the mentioned problem. The presented algorithm chooses the weighted path that starts from the up-left corner and reaches the down-right in the matrix of jobs processing times and calculates the biggest sum of the times in the footprints of this path. The row with the biggest sum permutes among all the rows of the matrix for locating the minimum of makespan. This method was run on Taillards standard benchmark and the solutions were compared with the optimum or the best ones as well as 14 famous heuristics. The validity and effectiveness of the algorithm are shown with tables and statistical evaluation. [ABSTRACT FROM AUTHOR]

Cruz, Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees, Appl. Math. Comput. 380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order. In this paper, we solve the open problem completely. [ABSTRACT FROM AUTHOR]

A kernel J of a digraph D is an independent set of vertices of D such that for every vertex w ϵ V (D) \ J there exists an arc from w to a vertex in J: In this paper, among other results, a characterization of 2-regular circulant digraph having a kernel is obtained. This characterization is a partial solution to the following problem: Characterize circulant digraphs which have kernels; it appeared in the book Digraphs - theory, algorithms and applications, Second Edition, Springer-Verlag, 2009, by J. Bang-Jensen and G. Gutin. [ABSTRACT FROM AUTHOR]