Mantel’s theorem is a classical result in extremal graph theory which implies that the maximum number of edges of a triangle-free graph of order n. In 1970, Nosal obtained a spectral version of Mantel’s theorem which gave the maximum spectral radius of a triangle-free graph of order n. In this paper, the clique tensor of a graph G is proposed and the spectral Mantel’s theorem is extended via the clique

For a purchase-to-order seller, there is no inventory and the seller has to purchase goods to fulfill orders already placed. For each purchase, there is a constant ordering cost. For each order, delay cost will be incurred if it is not fulfilled timely. Consequently, there is a tradeoff between the ordering cost and the delay cost for the seller to make replenishment decisions minimizing the total

In this paper, we consider several scheduling problems with rejection on \(m\ge 1\) identical machines. Each job is either accepted and processed on the machines, or it is rejected by paying a certain rejection cost. The objective is to minimize the sum of the k-th power of the makespan of accepted jobs and the total rejection cost of rejected jobs, where \(k>0\) is a given constant. We also introduce

We study a single machine lot scheduling problem. Customers’ orders may be processed simultaneously in the same lot. The order sizes are assumed to be order-dependent, and the lots have identical size and identical processing time. Orders may be split, i.e., their processing may be performed on two consecutive lots. We assume order-dependent due-dates and weights, and the goal is to minimize the maximum

The 0-1 linear programming problem with non-negative constraint matrix and objective vector \({\textbf{e}}\) origins from many NP-hard combinatorial optimization problems. In this paper, we consider under what condition an optimal solution of the 0-1 problem can be obtained from a weighted linear programming. To this end, we first formulate the 0-1 problem as a sparse minimization problem. Any optimal

In designing and understanding of computer networks, how to improve network robustness or protect a network from vulnerability remains an overarching concern. The residual closeness is a measure of network vulnerability and robustness even when the removal of vertices does not disconnect the underlying graph. We determine all the graphs that minimize and maximize the residual closeness respectively

The quadratic assignment problem (QAP) is perhaps the most widely studied nonlinear combinatorial optimization problem. It has many applications in various fields, yet has proven to be extremely difficult to solve. This difficulty has motivated researchers to identify special objective function structures that permit an optimal solution to be found efficiently. Previous work has shown that certain

For an additive submonoid \({\mathcal {M}}\) of \(\mathbb {R}_{\ge 0}\), the weight of a finite \({\mathcal {M}}\)-labeled directed graph is the sum of all of its edge labels, while the content is the product of the labels. Having fixed \({\mathcal {M}}\) and a directed tree E, we prove a general result on the shape of finite, acyclic, \({\mathcal {M}}\)-labeled directed graphs \(\Gamma \) of weight

We formulate two fairness principles and characterize the league competition systems that satisfy them. The first principle requires that all players should have the same chance of being the final winner if all players are equally strong, while the second states that the league competition should not favor weaker players. We apply these requirements to a class of systems which includes round-robin

This article studies the single-machine scheduling problem with due date assignments, deteriorating jobs, and past-sequence-dependent delivery times. Under three assignments (i.e., common, slack, and different due dates), the goal is to determine a feasible sequence and due dates of all jobs in order to minimize the weighted sum of earliness, tardiness, and due date costs of all jobs, where the weight

In this paper, we address the constrained parallel-machine scheduling problem with divisible processing times and penalties (the CPS-DTP problem), which is a further generalization of the parallel-machine scheduling problem with divisible processing times (the PS-DT problem). Concretely, given a set M of m identical machines and a set J of n independent jobs, each job has a processing time and a penalty