## 1 Introduction  
With its wide range of technical and scientific applications, the minimum spanning tree (MST) problem is one of the cornerstones of network optimization. The goal of the minimum spanning tree (MST) problem is to determine the least expensive minimally connected subgraph of a connected finite network. [PERSON] and [PERSON] [1] presented a version of the minimum spanning tree (MST) problem called the quadratic minimum spanning tree (QMST). This problem finds the minimum spanning tree of a weighted connected network by taking into account both linear and quadratic weights. Quadratic weights often result from the interaction effect between any two edges, while linear weights are attached to every edge individually. According to [PERSON] and [PERSON] [2], the QMST problem is useful in many areas of network design, including transportation networks, networks for the transmission of oil or water, and telecommunications networks. Usually, the linear weights in telecom networks often reflect the costs of setting up radio or cable connections, while the quadratic weights indicate the price of a device that converts between different types of linked cables. The cost of laying different pipes is represented by linear weights in a transportation or utility network, whereas the cost of interacting with things like valves or bending losses in T-joints is represented by quadratic weights in the same context. In a transportation network, the linear weights indicate the cost of developing roads, while the quadratic weights represent turn penalties [3]. QMST has been the subject of extensive investigation by several researchers over the past few decades. The branch-and-bound approach was first employed by [PERSON] and [PERSON] [1] to obtain the results. Subsequently, [PERSON] and [PERSON] [4] proposed an evolutionary approach for quality management system testing (QMST). The iterative tabu search method, hybrid genetic algorithm, and multi-start simulated annealing were presented by [PERSON] et al. [5] as potential solutions for the QMST problem. In 2010, [PERSON] and [PERSON] [6] employed the artificial bee colony algorithm that was designed in the form of a hive of bees to identify a solution. Afterwards in the same year, [PERSON] and [PERSON] [7] developed a local search method that is exceptionally efficient in solving the QMST problem. [PERSON] and [PERSON] [2] tackled the problem by introducing a tabu search and variable neighborhood search method. [PERSON] et al. [8] introduced an enhanced iteration of QMST capable of computing more robust linear programming bounds. In their study, [PERSON] et al. [9] proposed a methodology for addressing the QMST problem by integrating a tabu search with the strategic oscillation algorithm. The authors proposed a methodology that utilizes the creation of dynamic columns and rows in order to evaluate the bounds. In order to tackle this concern, [PERSON] and [PERSON] [10] employed a reformulation-linearization methodology to depict a range of novel mixed 0-1 linear constructs. [PERSON] et al. [8] subsequently tackled this problem by employing two parallel branch-and-bound algorithms, while also introducing an innovative formulation of the QMST. The study conducted by [PERSON] et al. [3] aimed to ascertain the existence of QMST and its three versions through an examination of NP-hard and polynomially solvable connected network problems. The three problems are: (i) the minimum spanning tree problem with conflict pair constraints, (ii) the bottleneck spanning tree problem with conflict pair constraints, and (iii) the quadratic bottleneck spanning tree problem. Finding a collection of node-disjoint cycles that visit all the nodes in a way that minimizes the total sum of interaction costs between subsequent arcs is the quadratic cycle cover problem that [PERSON] and [PERSON] [11] tackle in 2020. [PERSON] et al. [12] established the bottom bounds for the QMST using Semidefinite Programming (SDP) in the same year.