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An adjacency matrix A(G) of directed graph G is an m×m matrix consisting of only entries 0 and 1, where m is the number of vertices of G. The entry aij is equal to 1 if there exists a directed edge from vertex vi to vertex vj, otherwise it is equal to 0. Let D(G) be a diagonal matrix of size m×m with each of its main diagonal entry being the degree of the corresponding vertex of directed graph G. Then the matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G. Since a directed graph has two types of degrees namely indegree and outdegree, they result in directed graphs also having the both types of its Laplacian matrices. In this study, the adjacency matrix and the Laplacian matrix of cyclic directed prism graph are investigated. The general form of the coefficients of polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by applying the row reduction method in linear algebra, whereas the general form of the eigenvalues of the polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by factorization and substitution methods. [ABSTRACT FROM AUTHOR]

The energy of a graph is used to approximate the total π-electron energy of molecules. Thus, finding a new technique that facilitates this task is a challenge that has received increased research attention. This paper introduces the eigenvalues of a complete tripartite graph Ti,i,n-2i, for n ≥ 4, with respect to the adjacency, Laplacian, and signless Laplacian matrices. The chemical HMO approach is particularly successful in the case of the total π-electron energy. Experiments reveal that some chemical components are equienergetic with the tripartite graph. This finding helps easily derive the HMO of most of these components despite their different structures. [ABSTRACT FROM AUTHOR]

Graph theory was first introduced by Leonhard Euler in 1736 and still being one of the mathematic's topics which is rapidly developing and can be used to simplify mathematic problems. There are many interesting topics in graph theory; one of them is graph labeling. There are many ways of labeling a graph, and one of them is graceful labeling. Let G(V, E) is a graph. The injective mapping F ∶ V → {0, 1,…, |E|} is called graceful if the weights of edge w(uv) = |f(u) - f(v)| are all different for every edge uv. There is a famous conjecture in graceful labeling. It is said that all trees are graceful. To prove this conjecture, then we must show that every tree is graceful. There are many research papers dealing with the special cases of trees. Many classes of trees have been proven graceful, and one of them is supercaterpillar. Previous research had proved that supercaterpillars with certain conditions are also graceful. In this paper, we generalize the concept of supercaterpillar, and show that the subclass of supercaterpillar graphs that has not been discussed earlier is also graceful, using an adjacency matrix for the construction. [ABSTRACT FROM AUTHOR]

A graph is a set of vertices and edges where each edge connects two vertices in the graph. There are several ways to represent a graph, e.g., it can be represented as a matrix: its adjacency matrix, anti-adjacency matrix, Laplacian matrix, or distance matrix. Two matrices which will be explored in this paper are the adjacency matrix and distance matrix. The adjacency matrix represents the presence or absence of an edge connecting two vertices, while the distance matrix represents the shortest path between two vertices. A graph with diameter two is a graph such that the longest distance between any two vertices is equal to two. Examples of two-diameter graphs include bipartite graphs, wheel graphs, and fan graphs. The relation between the distance and adjacency matrices is already known. The purpose of this paper is to find the relation between the square of the distance matrix and the square of the adjacency matrix of a two diameter graph and to find the characteristic polynomial of the distance matrix of a complete bipartite graph Kn,n(which is a special type of two-diameter graph). We prove that D² = 4(J-I)Ā + A2 where D is the distance matrix, J is the matrix all of whose entries are 1, I is the identity matrix, A is the adjacency matrix and Ā is the complement of A Moreover, we also find that the characteristic polynomial of the distance matrix of a complete p bipartite graph is P(λ) = (λ+2)2n-2(λ-(n-2))(λ-(3n-2)). [ABSTRACT FROM AUTHOR]

The energy of a graph of a group G is the sum of all absolute values of the eigenvalues of the adjacency matrix. An adjacency matrix is a square matrix where the rows and columns consist of 0 or 1-entry depending on the adjacency of the vertices of the graph. A conjugacy class graph is a graph whose vertex set is the non-central conjugacy classes of the group. Two vertices are connected if their orders are not coprime. Meanwhile, a group G is said to be metabelian if there exists a normal subgroup H in G such that both H and the factor group G/H are abelian. In this research, the energy of the conjugacy class graphs for all nonabelian metabelian groups of order 24 are determined. The computations are assisted by Groups, Algorithm and Programming (GAP) and Maple2016 software. The results show that the energy of graphs of the groups in the study must be an even integer in the case that the energy is rational. [ABSTRACT FROM AUTHOR]

The traditional community detection method can only find the tightly connected structure, but the structure of the real network is hierarchical and diverse. Based on the minimum Description Length (MDL) principle, proposed a model consisting of the core node and 4 basic structures to represent the complex network and put forward a formula to calculate the minimum description length of network. First, delete the core in the network and get the hierarchical division, then decompose and merge the non-core network, and get the final network partition until the description length is minimal. Experimental results on real network dataset show that this method has better network compression effect, can get the hierarchical and diversified characteristics of the network, and get more realistic network partition. [ABSTRACT FROM AUTHOR]

In this work the evolution of visco-elastic systems under external stress is addressed. An approach as a mixed complementary eigenvalue problem to model the geological folding and asymmetric boudinage in the same direction is considered. A matricial dynamics equation that comprehends elasticity and viscosity matrices is presented. An algorithm to connect material points and to build the adjacency matrix has been developed. Numerical results for a set of 16 nodes are shown. [ABSTRACT FROM AUTHOR]

Circulating Fluidized Bed Boiler (CFB) is a device for generating steam by burning fossil fuels in a furnace operating under a special hydrodynamic condition. Autocatalytic Set has provided a graphical model of chemical reactions that occurred during combustion process in CFB. Eight important chemical substances known as species were represented as nodes and catalytic relationships between nodes are represented by the edges in the graph. In this paper, the model is extended and modified by considering other relevant chemical reactions that also exist during the process. Catalytic relationship among the species in the model is discussed. The result reveals that the modified model is able to gives more explanation of the relationship among the species during the process at initial time t. [ABSTRACT FROM AUTHOR]

Dry granular matter, with infinite tangential friction, is modelled as a graph of grains linked by purely repulsive contacts. Its stability (jamming) is insured by odd circuits that prevent the grains from rolling on each other. A topological dynamical matrix is associated with the graph; it has a spectrum of low-energy excitations characteristic of dry, disordered granular matter. In the limit of large stiffness-to load ratio, dry granular matter has two possible dynamical states, dry fluid and jammed, rigid but fragile solid. [ABSTRACT FROM AUTHOR]

Several recent advances in the theory of Zagreb indices are summarized. [ABSTRACT FROM AUTHOR]