For a finite group G and a nonempty subset X of G, we construct a graph with a set of vertex X such that any pair of distinct vertices of X are adjacent if they are commuting elements in G. This graph is known as the commuting graph of G on X, denoted by ΓG [X]. The degree exponent sum (DES) matrix of a graph is a square matrix whose (p,q)-th entry is is dvp dvq + dvqdvp whenever p is different from q, otherwise, it is zero, where dvp (or dvq ) is the degree of the vertex vp (or vertex, vq) of a graph. This study presents results for the DES energy of commuting graph for dihedral groups of order 2n, using the absolute eigenvalues of its DES matrix.

If G is a finite group and Z(G) is the centre of G, then the commuting graph for G, denoted by ΓG, has G\Z(G) as its vertices set with two distinct vertices vp and vq are adjacent if vp vq = vq vp. The degree of the vertex vp of ΓG, denoted by 𝑑𝑑𝑣𝑣𝑝𝑝 , is the number of vertices adjacent to vp. The maximum (or minimum) degree matrix of ΓG is a square matrix whose (p,q)-th entry is max{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 } (or min{𝑑𝑑𝑣𝑣𝑝𝑝,𝑑𝑑𝑣𝑣𝑞𝑞 }) whenever vp and vq are adjacent, otherwise, it is zero. This study presents the maximum and minimum degree energies of ΓG for dihedral groups of order 2n, D2n by using the absolute eigenvalues of the corresponding maximum degree matrices (MaxD(ΓG)) and minimum degree matrices (MinD(ΓG)). Here, the comparison of maximum and minimum degree energy of ΓG for D2n is discussed by considering odd and even n cases. The result shows that for each case, both energies are non-negative even integers and always equal.

For a finite group G, let Z(G) be the centre of G. Then the non-commuting graph on G, denoted by ΓG, has G\Z(G) as its vertex set with two distinct vertices vp and vq joined by an edge whenever vpvq ≠ vqvp. The degree sum matrix of a graph is a square matrix whose (p,q)-th entry is dvp + dvq whenever p is different from q, otherwise, it is zero, where dvi is the degree of the vertex vi. This study presents the general formula for the degree sum energy, EDS (ΓG), for the non-commuting graph of dihedral groups of order 2n, D2n, for all n ≥ 3.

PT. Sadar Jaya Manunggal merupakan salah satu perusahaan yang bergerak dalam pengadaan bahan bangunan. Perusahaan ini memiliki banyak cabang di kota-kota besar Indonesia, salah satunya di Kota Mataram yaitu di jalan TGH. Faisal no 78. Setiap hari, perusahaan akan melakukan pendistribusian bahan bangunan kepada para konsumen. Kegiatan pendistribusian ini memakan biaya dan waktu yang dipengaruhi oleh jarak setiap tempat yang menjadi tujuan pendistribusian, sehingga timbullah masalah bagaimana agar kegiatan pendistribusian ini  memakan biaya dan waktu seminimal mungkin, sehingga perusahaan memperoleh keuntungan yang optimal. Masalah tersebut merupakan bentuk travelling salesman problem yaitu mencari rute terpendek untuk pendistribusian bahan bangunan kepada semua konsumen.  Pemecahan permasalahan tersebut adalah dengan merepresentasikan peta tujuan pendistribusian atau alamat para konsumen ke dalam bentuk graf lengkap berbobot, selanjutnya permasalahan diselesaikan menggunakan Algoritma Branch and Bound. Perhitungan dilakukan secara manual dengan jarak (dalam kilometer) sebagai bobot perhitungan. Berdasarkan perhitungan menggunakan Algoritma Branch and Bound untuk optimasi rute pendistribusian bahan bangunan oleh PT. Sadar Jaya Manunggal Mataram menghasilkan solusi rute: (PT. Sadar Jaya Manunggal Mataram - UD. Mitra Utama - Pos Bangunan - Kunci Pelita - UD. Budi Rahman - Kurnia Jaya - UD. Salha - Ikhlas Bersama - PT. Sadar Jaya Manunggal Mataram) dengan total jarak 122,3 km.

The tight schedule of lecture activities requires accuracy so that it always runs smoothly. Lecturer assignments play an important role to ensure the smooth lecture activities. Problems that often occur in the assignment of these lecturers need to be avoided. In an effort to reduce the risk of problems that occur in the assignment of lecturers, it is necessary to make a structured system with the right method. Hungarian method can be said very appropriate for this assignment problem because each course will only be charged to one lecturer. Another advantage of using the hungarian method in this lecturer assignment model is also because it uses the preferences of prospective lecturers as subjects of measurement. Each lecturer will take courses according to their best preferences with the expectation that the lecturer will have more mastery in the courses that he will teach.

The neighbors degree sum (NDS) energy of a graph is determined by the sum of its absolute eigenvalues from its corresponding neighbors degree sum matrix. The non-diagonal entries of NDS−matrix are the summation of the degree of two adjacent vertices, or it is zero for non-adjacent vertices, whereas for the diagonal entries are the negative of the square of vertex degree. This study presents the formulas of neighbors degree sum energies of commuting and non-commuting graphs for dihedral groups of order 2n, D2n, for two cases−odd and even n. The results in this paper comply with the well known fact that energy of a graph is neither an odd integer nor a square root of an odd integer.

Eaglewood (Gyrinops verstiigee) is one example of a commercially valuable aromatic plant in the form of agarwood. The high selling value of agarwood encourages people to use it. The distribution area of Agarwood-producing trees is found in several regions in Indonesia. One of agarwood producers has been widely exploited in West Nusa Tenggara Province. The problem is the number of population and the quality of agarwood production which is declining due to exploitation carried out continuously and excessively without calculations and improper harvesting techniques. Therefore, a forest management model that cares for sustainability, called a sustainable harvesting model, could be used. This research used harvesting matrix to build this model. This model considers the initial configuration of the forest to be the same as the final configuration of the forest, so that the harvesting could be done continuously without damaging the forest configuration. The location of data collection was conducted in several areas, such as Orong Puncak, Lembah Sari, Orong Selatan, Kekait, Sepakek, East Lombok, and Kerujuk. The data obtained were divided into 5 intervals based on the diameter of the eaglewood tree. The research found that the number of trees that could be harvested on one harvesting period at each interval of trees diameter, i.e. 0 tree in 0-10 cm interval; 388 trees in 10-20 cm interval; 270 trees in 20-30 cm interval; 17 trees in 30-40 cm interval; and 3 trees in 40-50 cm interval. Harvesting matrix could be used to help the government in order to build a strategy for the sustainability of eaglewood trees in Lombok Island.

The kruskal algorithm is an algorithm to search for minimum spanning trees directly based on the general MST (Minimum Spanning Tree) algorithm. In the kruskal algorithm, the sides in the graph are sorted first based on their weight from small to large. The kruskal algorithm in the search for a minimum spanning tree can be applied to the distribution of clean water of PDAM in North Lombok district. This problem is intended to get the shortest route for PDAM water distribution in North Lombok district in order to minimize costs.