TRACE OF THE ADJACENCY MATRIX n×n OF THE CYCLE GRAPH TO THE POWER OF TWO TO FIVE (original) (raw)

The number of n-cycles in a graph

Applied Mathematics and Computation, 2007

Recently, Chang and Fu [Y.C. Chang, H.L. Fu, The number of 6-cycles in a graph, Bull. Inst. Combin. Appl. 39 (2003) 27–30] derived an exact expression, based on powers of the adjacency matrix, for the number of 6-cycles in a graph. Here, I demonstrate a method for obtaining the number of n-cycles in a graph from the immanants of the adjacency matrix. The method is applicable to cycles of all sizes and to sets of disjoint cycles of any sizes, and the cycles in the set need not be the same size.

On the Number of Cycles in a Graph

Open Journal of Discrete Mathematics

In this paper, we obtain explicit formulae for the number of 7-cycles and the total number of cycles of lengths 6 and 7 which contain a specific vertex v i in a simple graph G, in terms of the adjacency matrix and with the help of combinatorics.

Decomposition of complete tripartite graphs into 5-cycles using the graph adjacency matrix

2021

The problem of decomposing a complete tripartite graph into 5-cycles was first proposed in 1995 by Mahmoodian and Mirzakhani and since then many attempts have been made to decompose such graphs into 5-cycles. Such attempts were partially successful but parts of the problem still remain open. In this paper, we investigate the problem deeper, decompose more tripartite graphs into 5-cycles, and introduce the Graph Adjacency Matrix (GAM) method for cycle decomposition in general. GAM method transforms the cycle decomposition problem to covering squares with certain polygons. This new formulation is easier to solve and enables us to find explicit decompositions for numerous cases that were not decomposed before.

The adjacency matrix of one type of graph and the Fibonacci numbers

2012

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we investigate relationships between one type of graph and well-known Fibonacci sequence. In this content, we consider the adjacency matrix of one type of graph with 2k (k = 1, 2, ...) vertices. It is also known that for any positive integer r, the (i, j)th entry of A r (A is the adjacency matrix of the graph) is just the number of walks from vertex i to vertex j, that use exactly k edges.

Cycle bases in graphs characterization, algorithms, complexity, and applications

Computer Science Review, 2009

Cycles in graphs play an important role in many applications, e.g., analysis of electrical networks, analysis of chemical and biological pathways, periodic scheduling, and graph drawing. From a mathematical point of view, cycles in graphs have a rich structure. Cycle bases are a compact description of the set of all cycles of a graph. In this paper, we survey the state of knowledge on cycle bases and also derive some new results. We introduce different kinds of cycle bases, characterize them in terms of their cycle matrix, and prove structural results and apriori length bounds. We provide polynomial algorithms for the minimum cycle basis problem for some of the classes and prove APX -hardness for others. We also discuss three applications and show that they require different kinds of cycle bases.

A study on domatic number of cycle related graphs

Journal of Ambient Intelligence and Humanized Computing, 2020

A domatic partition of G is the partition of vertices V(G) into disjoint dominating sets. The maximum size of disjoint dominating sets is called the domatic number of G. In this paper, comparative results are investigated on domatic partition of few graphs of cycle related graphs such as complete graph, tadpole graph, lollipop graph and barbell graph for cognitive wireless sensor networks. Then the middle graph and central graph of these graphs are studied and domatic number of the defined graphs are determined to find the nodes in disjoint sets to disribute the tasks uniformly rather than burden the nodes in domatic set. Furthermore, diameter and domination number of these graphs are observed.

The characteristic polynomial of a graph

Journal of Combinatorial Theory, Series B, 1972

The present paper is addressed to the problem of determining under what conditions the characteristic polynomial of the adjacency matrix of a graph distinguishes between non-isomorphic graphs. A formula for the coeiiicients of the characteristic polynomial of an arbitrary digraph is derived, and the polynomial of a tree is examined in depth. It is shown that the coefFicients of the polynomial of a tree count matchings. Several recurrence relations are also given for computing the coefficients. An appendix is provided which lists n-node trees (2 < N < 10) together with the coefficients of their polynomials. It should be aoted that this list corrects some errors in the earlier table of [I].

The Cycle-Path Indicator Polynomial of a Digraph

Advances in Applied Mathematics, 2000

A cycle-path cover of a digraph D is a spanning subgraph made of disjoint cycles and paths. In order to count such covers by types we introduce the cyclepath indicator polynomial of D. We show that this polynomial can be obtained by a deletionxontraction recurrence relation. Then we study some specializations of the cycle-path indicator polynomial, such as the geomettic coverpo&nomial (the geometric version of the cover polynomial introduced by Chung and Graham), the cycle cover polynomial, and the path cover polynomial.

Graphs with circulant adjacency matrices

Journal of Combinatorial Theory, 1970

Properties of a graph (directed or undirected) whose adjacency matrix is a circulant are studied. Examples are given showing that the connection set determined by the first row of such a matrix need not be multiplicatively related to the connection set of an isomorphic graph. Two different criteria are given under which two graphs with circulant adjacency matrices are isomorphic if and only if their connection sets are multiplicatively related. The first criterion is that the graphs have a prime number of vertices. The second criterion is that the adjacency matrices have non-repeated eigenvalues. The final section gives a partial characterization of graphs with n vertices whose automorphism group is the cyclic group C~.