For an integer k >= 0, a connected graph G is called a k-cactus graph if each edge e is an element of E(G) is contained in at most k cycles of G. Inspired by the Brualdi-Solheid problem, in this paper, we address the problem of determining the maximum spectral radius of k-cactus graphs. Lovasz and Pelikan [Period. Math. Hungar.3 (1973) 175-182], Borovicanin and Petrovic [Publ. Inst. Math. (Beograd) (N.S.)79 (2006) 13-18] resolved the cases of k = 0 and k = 1, respectively. We solve this problem for the cases of k = 2 and k = 3 that is the graphs with the maximum spectral radius among all 2-cactus graphs and 3-cactus graphs are determined, respectively.
We study a mathematical program with switching constraints (MPSC). By introducing a suitable constraint qualification and an alternative stationarity concept, we derive necessary optimality conditions using D-directional upper convexificators and D-directional upper semi-regular convexificators, where D is a nonempty closed cone within the set of continuity directions for functions that may lack continuity. An illustrative example is provided.
Minimizing maximum cost and makespan simultaneously on a bounded series-batching machine is considered in the paper, in which each job contains a specific component and a standard component. Specific components are scheduled separately, while standard components are scheduled in batches. Completing a job means that its two components have been completed. Additionally, the two components nents of a job can be scheduled in any order. We present an O ( n 4 )-time algorithm for the simultaneous optimization scheduling problem. When the maximum cost is the maximum lateness, time complexity of the algorithm is O ( n 3 log n ) time.
The goal of this article is to obtain closed formulas for the outer-independent total domina- tion number and the outer-independent total Roman domination number of the following well-known graph operators defined from a connected graph G : the central graph C(G) the middle graph M(G) the graph R(G) and the total graph T(G) . We show that these formulas can be obtained for the last three graph operators and for the outer-independent total domination number of C(G) . The picture is quite different when it concerns the outer-independent total Roman domination number of C(G) . In this case, we obtain tight bounds and, imposing some restrictions on G , we obtain closed formulas.
This study examines the steady-state dynamics of an M/G /1 retrial queue that incorpo- rates features such as balking, optional service, negative customer, and general retrial time under a constant retrial policy. The probability generating functions for the orbit length with server status are obtained using the supplementary variable technique at various time epochs. We further extract the orbit size probabilities from these generating functions using the roots method. Additionally, the tail distribution of the orbit length is also presented for different time epochs. We conduct a comprehensive cost analysis to minimize the overall system operation cost. Advanced metaheuristic techniques such as particle swarm optimization and genetic algorithm are employed to determine the optimal system parameters that minimize the total cost. The impact of various system parameters on performance measures is demonstrated through detailed numerical examples.
