Antiadjacency Matrix Research Articles

Several properties of antiadjacency matrices of directed graphs

<p>Let $ G $ be a directed graph with $ \mathrm{o}\mathrm{r}\mathrm{d}\mathrm{e}\mathrm{r}\; n. $ The adjacency matrix of the directed graph $ G $ is a matrix $ A = \left[{a}_{ij}\right] $ of order $ n\times n, $ such that for $ i\ne j $, if there is an arc from $ i $ to $ j, $ then $ {a}_{ij} = 1 $, otherwise $ {a}_{ij} = 0 $. Matrix $ B = J-A $ is called the antiadjacency matrix of the directed graph $ G, $ where $ J $ is the matrix of order $ n\times n $ with all of those entries are one. In this paper, we provided several properties of the adjacency matrices of directed graphs, such as a determinant of a directed graphs, the characteristic polynomial of acyclic directed graphs, and regular directed graphs. Moreover, we discuss antiadjacency energy of acyclic directed graphs and give some examples of antiadjacency energy for several families of graphs.</p>

ANTIADJACENCY MATRICES FOR SOME STRONG PRODUCTS OF GRAPHS

Let G be an undirected graphs with no multiple edges. There are many ways to represent a graph, and one of them is in a matrix form, by constructing an antiadjacency matrix. Given a connected graph G with vertex set $V$ consisting of n members, an antiadjacency matrix of the graph G is a matrix B of order n \times n such that if there is an edge that connects vertex v_i to vertex v_j (v_i \sim v_j ) then the element of i^{th} row and b^{th} column of B is 0, otherwise 1. In this paper we investigate some properties of antiadjacency matrices for some strong product of two graphs. Our results are general forms of the antiadjacency matrix of the strong product of path graphs P_m with P_n for m, n\ge 3, and cycle graphs C_m with C_m for m \ge 3.

On two problems related to anti-adjacency (eccentricity) matrix

On two problems related to anti-adjacency (eccentricity) matrix

On graphs with exactly one anti-adjacency eigenvalue and beyond

The anti-adjacency matrix of a graph is constructed from the distance matrix of a graph by keeping each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping in each row and each column only the distances equal to 1. The (anti-)adjacency eigenvalues of a graph are those of its (anti-)adjacency matrix. Employing a novel technique introduced by Haemers (2019) [9], we characterize all connected graphs with exactly one positive anti-adjacency eigenvalue, which is an analog of Smith's classical result that a connected graph has exactly one positive adjacency eigenvalue iff it is a complete multipartite graph. On this basis, we identify the connected graphs with all but at most two anti-adjacency eigenvalues equal to −2 and 0. Moreover, for the anti-adjacency matrix we determine the HL-index of graphs with exactly one positive anti-adjacency eigenvalue, where the HL-index measures how large in absolute value may be the median eigenvalues of a graph. We finally propose some problems for further study.

Eigenvalues of antiadjacency matrix of Cayley graph of Z_n

<p>In this paper, we give a relation between the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) and the eigenvalues of the antiadjacency matrix of Cay(Z_n, (Z_n−{0})−S), as well as the eigenvalues of the adjacency matrix of Cay(Z_n, S). Then, we give the characterization of connection set S where the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) are all integers.</p>

ON ANTIADJACENCY MATRIX OF A DIGRAPH WITH DIRECTED DIGON(S)

The antiadjacency matrix is one representation matrix of a digraph. In this paper, we find the determinant and the characteristic polynomial of the antiadjacency matrix of a digraph with directed digon(s). The digraph that we will discuss is a digraph obtained by adding arc(s) in an arborescence path digraph such that it contained directed digon(s), and a digraph obtained by deleting arc(s) in a complete star digraph. We found that the determinant and the coefficient of the characteristic polynomial of the antiadjacency matrix of a digraph obtained by adding arc(s) in an arborescence path digraph such that it contained directed digon(s) is different depending on the location of the directed digon. Meanwhile, the determinant of the antiadjacency matrix of a digraph obtained by deleting arc(s) in the complete star digraph is zero.

On Characteristic Polynomial of Antiadjacency Matrix of A Line Digraph

In this paper, we find the characteristic polynomial of the antiadjacency matrix of a line digraph. There are recent studies on the relation between the characteristic polynomial of the adjacency matrix and its line digraph, we are also interested in finding the connection between the antiadjacency matrix of a digraph and its line digraph. In this paper, we show the connection of characteristic polynomial of the antiadjacency matrix between an acyclic digraph and its line digraph.

CHARACTERISTIC ANTIADJACENCY MATRIX OF GRAPH JOIN

Let be a simple, connected, and undirected graph. The graph can be represented as a matrix such as antiadjacency matrix. An antiadjacency matrix for an undirected graph with order is a matrix that has an order and symmetric so that the antiadjacency matrix has a determinant and characteristic polynomial. In this paper, we discuss the properties of antiadjacency matrix of a graph join, such as its determinant and characteristic polynomial. A graph join is obtained of a graph join operation obtained from joining two disjoint graphs and .

Spectral determination of graphs with one positive anti-adjacency eigenvalue

The anti-adjacency matrix (or eccentricity matrix) of a graph is obtained from its distance matrix by retaining for each row and each column only the largest distances. This matrix can be viewed as the opposite of the adjacency matrix, which is, on the contrary, obtained from the distance matrix of a graph by keeping for each row and each column only the distances being 1. In this paper, we prove that the graphs with exactly one positive anti-adjacency eigenvalue are determined by the anti-adjacency spectra. As corollaries, the well-known (generalized) friendship graphs and windmill graphs are shown to be determined by their anti-adjacency spectra.

Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph

A directed cyclic graph is a directed graph that has at least one directed cycle graph, that is a cycle graph in which all edges are oriented, so that the direction passes through each vertex once, except the end of vertex. The directed unicyclic tadpole graph is the graph created by concatenating a cycle and a path with an edge from any vertex of to a pendant of for integers m ≥ 3 and n ≥ 1. Antiadjacency matrix is a directed graph representation matrix based on whether or not there is a relation between one vertex and the others. This paper gives the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph. To find the general form of coefficients of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic tadpole graph we need to do the grouping of induced subgraphs into acyclic and cyclic and verify with related theorems. After that, the characteristic polynomial is factorized and the roots are calculatedto find its eigenvalues. The coefficients of the characteristic polynomial consist of three distinct values and the eigenvalues are divided into odd case and even case.

Characteristic polynomial and eigenvalues of anti-adjacency matrix of directed unicyclic corona graph

A directed graph can be represented by several matrix representations, such as the anti-adjacency matrix. This paper discusses the general form of characteristic polynomial and eigenvaluesof the anti-adjacencymatrix of directed unicyclic corona graph. The characteristic polynomial of the anti-adjacency matrix can be found by counting the sum of the determinant of the anti-adjacency matrix of the directed cyclic inducedsubgraphs and the directed acyclic induced subgraphs from the graph. The eigenvalues of the anti-adjacency matrix can be real or complex numbers. We prove that the coefficient of the characteristic polynomial and the eigenvalues of the anti-adjacency matrix of directed unicyclic corona graph can be expressed in the function form that depends on the number of subgraphs contained in thedirected unicyclic corona graphs.

Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic helm graph

A directed unicyclic graph is a directed graph that has only one directed cycle subgraph. A directed unicyclic helm graph is obtained from a directed wheel graph by adjoining a directed pendant edge at each vertex of the cycle. A directed graph can be represented into several matrix representations, one of them is the antiadjacency matrix. The antiadjacency matrix is a matrix in which the entries represent whether there is a directed edge from one vertex to another. This paper discusses the general form of the coefficients of the characteristic polynomial that obtained by adding all of the determinants of antiadjacency matrix from each induced acyclic and cyclic subgraphs. The eigenvalues of the antiadjacency matrix of the directed unicyclic helm graph obtained by polynomial factorization. The result obtained denotes that the coefficients of the characteristic polynomial and eigenvalues of the antiadjacency matrix depend on the number of vertices of the cycle subgraphs of directed unicyclic helm graph.

Characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph

This research discussed the characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The entries of the antiadjacency matrix of a directed graph represent the presence or the absence of a directed arc from one vertex to the others. If A is the adjacency matrix of a graph , then the antiadjacency matrix B of graph is B = J − A, where J is the square matrix with all entries equal to one. The general form of characteristic polynomial coefficients of the antiadjacency matrix of directed unicyclic flower vase graph can be obtained by calculating the sum of determinants of the antiadjacency matrices of all induced cyclic and acyclic subgraphs, while the eigenvalues were obtained by using polynomial factorization and Horner’s method. This paper gives the characteristic polynomial coefficients and eigenvalues of antiadjacency matrix of directed unicyclic flower vase graph. The characteristic polynomial can be considered as a function that depends on the number of vertices.

Characteristic polynomial of anti-adjacency matrix of directed cyclic friendship graph

Graph theory has some applications. One of them is used to do social network analysis as in Facebook with each person as nodes and every like, share, comment, tag as edges. Usually a network can be represented by a graph. Analysing a network is the same as analysing the structure of a graph. A structure of a graph can be analysed through some matrix representations such as anti-adjacency matrix. The entries of the anti-adjacency matrix of a directed graph represent the presence or absence of directed arcs from a vertex to the others. This paper is about the properties of the anti-adjacency matrix of directed cyclic friendship graph, those are the characteristic polynomial and the eigenvalues of the anti-adjacency matrix. The method used to obtain the general form of the coefficients of the characteristic polynomial is by adding up the determinant values of all directed induced subgraphs, cyclic or acyclic, of the directed cyclic friendship graph. Furthermore, the methods used to find theeigenvalues of the anti-adjacency matrix of directed cyclic friendship graph are the substitution method and factorization method. The result of this researcharethe general form of the coefficients of the characteristic polynomial and the general form of the eigenvalues of the anti-adjacency matrix depend on the number of triangles in directed cyclic friendship graph.

Properties of anti-adjacency matrix of directed cyclic sun graph

In this paper we focus on the properties of anti-adjacency matrix of directed cyclic sun graph. Some of these properties are related to the characteristic polynomials and the eigenvalues of the anti-adjacency of its matrix. We will show the general form of characteristic polynomial of the anti-adjacency matrix of directed cyclic sun graph by figuring out the number of the directed induced-cyclic graphs and the directed induced-acyclic graphs. After we find out the general form of the characteristic polynomial, we can find the general form of the eigenvalues of its polynomial by using factorization and Horner methods.

Finite-Time Stability of Probabilistic Logical Networks: A Topological Sorting Approach

This brief presents some further results on the finite-time stability of probabilistic logical networks (PLNs). By semi-tensor product technique routinely, the dynamic behavior of a PLN is characterized by its corresponding state transition graph (STG). Then, an irradiative result is found. That is, a PLN is globally stable within finite time, if and only if, its STG is acyclic, except for the self loop at the pre-designated vertex. Based on this observation, some properties of STG, which is associated with a finite-time stable PLN, are formulated. The most significant finding is that the determinant of its anti-adjacency matrix is compactly related to the existence of a Hamilton path and is only equal to 0 or 1. Afterwards, the topological sort of all the vertices in STG is defined. As a consequence, two topological sorting algorithms are presented to analyze the stability of PLNs applicably and efficiently. Finally, a simulation example is employed to illustrate the applicability of the obtained results.

The adjacency matrix of directed cyclic wheel graph

An adjacency matrix is one of the matrix representations of a directed graph. In this paper, the adjacency matrix of a directed cyclic wheel graph is denoted by . From the matrix the general form of the characteristic polynomial and the eigenvalues of a directed cyclic wheel graph can be obtained. The norm of each coefficient of its characteristic polynomial is obtained by calculating the sum of the principal minors of subgraphs of the graph . It turns out that each coefficient ai, i = 1, 2, …, (n − 2), n equals zero except an−1 is equal to -1. In addition the matrix has real eigenvalues and also some complex eigenvalues that conjugate each other. The real eigenvalues are obtained by searching the real roots through the characteristic equation, then by factorization we get the polynomial factor that contains the complex roots. The complex eigenvalues of the adjacency matrix of a directed cyclic wheel graph or has a relation with the complex eigenvalues of the antiadjacency matrix of a directed cyclic wheel graph or , that is the complex eigenvalues of the matrix are equal to the negative of the complex eigenvalues of the matrix .

Characteristic Polynomial of Antiadjacency Matrix of Directed Cyclic Wheel Graph

A directed cyclic wheel graph with order n, where n ≥ 4 can be represented by an anti-adjacency matrix. The anti-adjacency matrix is a square matrix that has entries only 0 and 1. The number 0 denotes an edge that connects two vertices, whereas the number 1 denotes otherwise. The norm of every coefficient in characteristic polynomial of the anti-adjacency matrix of a directed cyclic wheel graph represents the number of Hamiltonian paths contained in the induced sub-graphs minus the number of the cyclic induced sub-graphs. In addition, the eigenvalues can be found through the anti-adjacency matrix of directed cyclic wheel graph. The result is, the anti-adjacency matrix of directed cyclic wheel graph has two real eigenvalues and some complex eigenvalues that conjugate to each other. The real eigenvalues are obtained by Horner method, while the complex eigenvalues are obtained by finding the complex roots from the factorization of the characteristic polynomial.

Eigenvalues of Antiadjacency Matrix of Directed Cyclic Dumbbell Graph

This paper explains the steps used to find the characteristic polynomial of the antiadjacency matrix of a cyclic dumbbell graph. The antiadjacency matrix of a graph is a matrix whose entries represent whether there is an edge that connects two vertices or not. The general form of the characteristic polynomial includes its eigenvalues. The antiadjacency matrix is obtained by using some theorems, the number of solutions of integer equations, quadratic formula, and polynomials factorization. Finally, results showed that the coefficients of the characteristic polynomial and its eigenvalues were dependent on the number of vertices of the cyclic dumbbell graph.

The anti-adjacency matrix of a graph: Eccentricity matrix

In this paper we introduce a new graph matrix, named the anti-adjacency matrix or eccentricity matrix, which is constructed from the distance matrix of a graph by keeping for each row and each column only the largest distances. This matrix can be interpreted as the opposite of the adjacency matrix, which is instead constructed from the distance matrix of a graph by keeping for each row and each column only the distances equal to 1. We show that the eccentricity matrix of trees is irreducible, and we investigate the relations between the eigenvalues of the adjacency and eccentricity matrices. Finally, we give some applications of this new matrix in terms of molecular descriptors, and we conclude by proposing some further research problems.