Characteristic Polynomial Of Graph Research Articles

Characteristic Polynomial of Power Graph for Dihedral Groups Using Degree-Based Matrices

A fundamental feature of spectral graph theory is the correspondence between matrix and graph. As a result of this relation, the characteristic polynomial of the graph can be formulated. This research focuses on the power graph of dihedral groups using degree-based matrices. Throughout this paper, we formulate the characteristic polynomial of the power graph of dihedral groups based on seven types of graph matrices which include the maximum degree, the minimum degree, the greatest common divisor degree, the first Zagreb, the second Zagreb, the misbalance degree, and the Nirmala matrices.

Spectral Invariants and Their Application on Spectral Characterization of Graphs

In this paper, we give a method to characterize graphs determined by their adjacency spectrum. At first, we give two parameters Π1(G) and Π2(G), which are related to coefficients of the characteristic polynomial of graph G. All connected graphs with Π1(G)∈{1,0,−1,−2,−3} and Π2(G)∈{0,−1,−2,−3} are characterized. Some interesting properties of Π1(G) and Π2(G) are also given. We then find the necessary and sufficient conditions for two classes of graphs to be determined by their adjacency spectrum.

A new method for the sum-edge characteristic polynomials of graphs

In this paper, the determinant of the sum-edge adjacency matrix of any given graph without loops is calculated by means of an algebraic method using spanning elementary subgraphs and also the coefficients of the corresponding sum-edge characteristic polynomial are determined by means of the elementary subgraphs. Also we gave a formula for the number of smallest odd-sized cycles in a given regular graph.

The characteristic polynomial of a graph containing loops

We focus on the characteristic polynomial of a graph containing loops, but without multiple edges. We present a relationship between the characteristic polynomial of a graph with loops and the graph obtained by removing all the loops. In turn, we compute the characteristic polynomial of unitary addition Cayley graphs.

Embedding of sign-regular signed graphs and its spectral analysis

ABSTRACT A is a graph whose edges carry the weight ‘+’ or ‘−’. A signed graph S is called sign-regular if is same for all and is same for all . The problems of embedding -sign-regular signed graphs in -sign-regular signed graphs is one of the fascinating problem from application point of view which is dealt in this paper with insertion of least number of vertices in S and the problem of finding least number of non-isomorphic co-regular signed graphs is explored. We also define the relationship between characteristic polynomial of graph G and the graph in which it embeds.

Sum-edge characteristic polynomials of graphs

Modelling a chemical compound by a (molecular) graph helps us to obtain some required information about the chemical and physical properties of the corresponding molecular structure. Linear algebraic notions and methods are used to obtain several properties of graphs usually by the help of some graph matrices and these studies form an important sub area of Graph Theory called spectral graph theory. In this paper, we deal with the sum-edge matrices defined by means of vertex degrees. We calculate the sum-edge characteristic polynomials of several important graph classes by means of the corresponding sum-edge matrices.

Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs II

Let G be a graph of order n and λ1≥λ2≥⋯≥λn be the eigenvalues of G. The energy of G is defined as E(G)=∑k=1n|λk|. A well-known result regarding the energy of graphs is the Coulson integral formula, which defines the relationship between the energy and the characteristic polynomial of graphs. Let μ1≥μ2≥⋯≥μn=0 be the Laplacian eigenvalues of G. The general Laplacian energy-like invariant of G, denoted by LELα(G), is defined as ∑μk≠0μkα when μ1≠0, and 0 when μ1=0, where α is a real number. In this study, we give some Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs in the case where α is a rational number. Based on this result, we also give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case where α is a rational number. We also show that our formulas hold when α is an irrational number where 0<|α|<1, whereas they do not hold when |α|>1.

Coulson-type integral formulas for the general Laplacian-energy-like invariant of graphs I

Let G be a simple graph. Its energy is defined as E(G)=∑k=1n|λk|, where λ1,λ2,…,λn are the eigenvalues of G. A well-known result on the energy of graphs is the Coulson integral formula which gives a relationship between the energy and the characteristic polynomial of graphs. Let μ1≥μ2≥⋯≥μn=0 be the Laplacian eigenvalues of G. The general Laplacian-energy-like invariant of G, denoted by LELα(G), is defined as ∑μk≠0μkα when μ1≠0, and 0 when μ1=0, where α is a real number. In this paper we give a Coulson-type integral formula for the general Laplacian-energy-like invariant for α=1/p with p∈Z+\\{1}. This implies integral formulas for the Laplacian-energy-like invariant, the normalized incidence energy and the Laplacian incidence energy of graphs.

The New Class of Laplacian Integral Graphs

Graph is Laplacian integral, if all the eigenvalues of its Laplacian matrix are integral. In this paper, we obtain that the Laplacian characteristic polynomials of graphs by calculation. Characterizes the new class of Laplacian integral graphs .

On the log-concavity of Laplacian characteristic polynomials of graphs

Let G be a graph and L(G) be the Laplacian matrix of G. In this article, we first point out that the sequence of the moduli of Laplacian coefficients ofG is log-concave and hence unimodal. Using this fact, we provide an upper bound for the partial sums of the Laplacian eigenvalues of G, based on coefficients of its Laplacian characteristic polynomial. We then obtain some lower bounds on the algebraic connectivity of G. Finally, we investigate the mode of such sequences. 1. Introduction. Throughout this paper, we consider simple undirected graphs having n vertices and m edges. For a given graph G, let V (G) and E(G) denote the vertex and the edge set of G, respectively. Let A(G) be the adjacency matrix of G. The Laplacian matrix and signless Laplacian matrix of G are defined as L(G) = D(G) − A(G) and Q(G) = D(G)+A(G), respectively, where D(G) is a diagonal matrix whose diagonal entries are vertex degrees of G. The eigenvalues of matrices L(G) and Q(G) are denoted by µ1(G) ≥ µ2(G) ≥ ··· ≥ µn(G) = 0 and �1(G) ≥ �2(G) ≥ ··· ≥ �n(G), respectively. As it is well-known, L(G) and Q(G) are positive semi-definite, and they have the same characteristic polynomial if and only if the graph G is bipartite. The second smallest eigenvalue of L(G), µn−1(G), is called the algebraic connectivity of G, and it is positive if and only if the graph G is connected. For bibliographies on the graph Laplacian, the reader is referred to (11).

Another form of the transmission function

The transmission function describes the passage of the electric current from one point of an electric circuit to another. By now, this is also applied to molecules which are potential candidates for uses in the molecular electronics. We mean the modern branch of electronics which has a goal of reducing the sizes of its devices down to molecular ones and planning indeed to apply single molecules as conducting wires and functional components of microcircuits. For calculating the transmission function, some authors utilize the well-known idea of representing a molecule by a (molecular) graph, which allows them to apply for treating the latter also powerful methods of spectral graph theory. For instance, we refer to the paper by Fowler et al. (Chem Phys Lett. 465 2008) 142–146, where one such expression for this function is given. Our objective is to demonstrate that the same calculational result can be obtained using a different set of characteristic polynomials of graphs (which also slightly reduces a mathematical notation). Specifically, we apply one theorem of Kolmykov to the basic formula derived by these authors.

Signless Laplacian Polynomial and Characteristic Polynomial of a Graph

The signless Laplacian polynomial of a graph G is the characteristic polynomial of the matrix Q(G)=D(G)+A(G), where D(G) is the diagonal degree matrix and A(G) is the adjacency matrix of G. In this paper we express the signless Laplacian polynomial in terms of the characteristic polynomial of the induced subgraphs, and, for regular graph, the signless Laplacian polynomial is expressed in terms of the derivatives of the characteristic polynomial. Using this we obtain the characteristic polynomial of line graph and subdivision graph in terms of the characteristic polynomial of induced subgraphs.

The conjugate characteristic polynomial of some compound graphs

Let G be a simple graph of order n and let S be any subset of the vertex set V(G). We denote by G S, T the graph obtained from G by adding two new non-adjacent vertices x, y, so that x is adjacent exactly to the vertices from S, and y is adjacent exactly to the vertices from T, where T=V(G)∖S. We here determine the conjugate characteristic polynomial of the graph G S, T . We also consider some other compound graphs obtained from G and determine their conjugate characteristic polynomials.

Characteristic polynomials of digraphs having a semi-free action

Let ϕ be a semi-free action of a group G on a finite digraph Γ. The front divisor Γ/ π( ϕ) of Γ has as vertex set the set of the vertex orbits of ϕ and there are t arcs going from u ¯ to v ¯ in Γ/ π( ϕ) if from any vertex u of the orbit u ¯ there are t arcs of Γ going towards the vertices in v ¯ . Our main result is that the characteristic polynomial of Γ is a product of the characteristic polynomial of its front divisor Γ/ π( ϕ) and polynomials associated with the free part of Γ under ϕ. This work extends earlier work of Lee and Kim [Characteristic polynomials of graphs having a semifree action, Linear Algebra Appl. 307 (2000) 35–46].

Characteristic polynomials of graphs having a semifree action

J.H. Kwak and J. Lee (Linear and Multilinear Algebra 32 (1992) 61–73) computed the characteristic polynomial of a finite graph G having an abelian automorphism group which acts freely on G. For a finite weighted symmetric pseudograph G having an abelian automorphism group which acts semifreely on G, K. Wang (Linear Algebra Appl. 51 (1983) 121–125) showed that the characteristic polynomial of G is factorized into a product of a polynomial associated to the orbit graph and a polynomial associated to the free part of the action. But he did not explicitly compute the characteristic polynomial of such a graph G. In this paper, we introduce a new method to construct a finite pseudograph G having an automorphism group which acts semifreely on G, and obtain an explicit formula to compute the characteristic polynomial of such a graph by using the construction method.

The characteristic polynomial of a graph is reconstructible from the characteristic polynomials of its vertex-deleted subgraphs and their complements

The question of whether the characteristic polynomial of a simple graph is uniquely determined by the characteristic polynomials of its vertex-deleted subgraphs is one of the many unresolved problems in graph reconstruction. In this paper we prove that the characteristic polynomial of a graph is reconstructible from the characteristic polynomials of the vertex-deleted subgraphs of the graph and its complement.

Characteristic polynomials and spectra of Boolean graphs

Three theorems are proved by using fundamental concepts concerned with the eigenvectors and the dimension of the space of eigenvectors and by considering that the Boolean graph Bn is a regular graph of nth degree. The results are discussed by applying these theorems to graphs B 1, B 2, B 3 .It is shown that the positive integer nis the greatest eigenvalue of Bn so that multiplicity of n is one and the negative integer — n is the smallest eigenvalue of Bn so that multiplicity of — nis one. Hence, by making a suitable generalization to the spectrums and characteristic polynomials of graphs B 1, B 2, B 3 .general formulas are presented related with the discovery of all spectrums and characteristic polynomials of graphs Bn (n ϵ Z+).

On the index of broken wheels

The index of a graph G is the largest eigenvalue of a (0,1)-adjacency matrix of G. Let denote the set of all graphs which can be obtained from an n-cycle by joining and additional ('central') vertex to k of the vertices of the cycle (Such graphs are called broken wheels.) By using a result of Schwenk's to compare the characteristic polynomials of graphs in . we identify the graphs with greatest and least index. We show in fact that the index is greatest when the k 'spokes' are bunched together as closely as possible, and is least when they are spread out as evenly as possible.

On the characteristic polynomials of spirographs and related graphs

General recursive techniques are used to determine recurrence relations for the characteristic polynomials of graphs associated with various ring compounds.

The use of Frame’s method for the characteristic polynomials of chemical graphs

The evaluation of characteristic polynomials of graphs of any size is an extremely tedious problem because of the combinatorial complexity involved in this problem. While particular elegant methods have been outlined for this problem, a general technique for any graph is usually tedious. We show in this paper that the Frame method for the characteristic polynomial of a matrix is extremely useful and can be applied to graphs containing large numbers of vertices. This method reduces the difficult problem of evaluating the characteristic polynomials to a rather simple problem of matrix products. The coefficients in the characteristic polynomial are generated as traces of matrices generated in a recursive product of two matrices. This method provides for an excellent and a very efficient algorithm for computer evaluation of characteristic polynomials of graphs containing a large number of vertices without having to expand the secular determinant of the matrix associated with the graph. The characteristic polynomials of a number of graphs including that of a square lattice containing 36 vertices are obtained for the first time.