The characteristic polynomial of a graph (original) (raw)

Matching number and characteristic polynomial of a graph

Journal of Taibah University for Science

Matching number and the spectral properties depending on the characteristic polynomial of a graph obtained by means of the adjacency polynomial has many interesting applications in different areas of science. There are some work giving the relation of these two areas. Here the relations between these two notions are considered and several general results giving this relations are obtained. A result given for only unicyclic graphs is generalized. There are some methods for determining the matching number of a graph in literature. Usually nullity, spanning trees and several graph parts are used to do this. Here, as a new method, the conditions for calculating the matching number of a graph by means of the coefficients of the characteristic polynomial of the graph are determined. Finally some results on the matching number of graphs are obtained.

QAPV: a polynomial invariant for graph isomorphism testing

Pesquisa Operacional, 2013

To each instance of the Quadratic Assignment Problem (QAP) a relaxed instance can be associated. Both variances of their solution values can be calculated in polynomial time. The graph isomorphism problem (GIP) can be modeled as a QAP, associating its pair of data matrices with a pair of graphs of the same order and size. We look for invariant edge weight functions for the graphs composing the instances in order to try to find quantitative differences between variances that could be associated with the absence of isomorphism. This technique is sensitive enough to show the effect of a single edge exchange between two regular graphs of up to 3,000 vertices and 300,000 edges with degrees up to 200. Planar graph pairs from a dense family up to 300,000 vertices were also discriminated. We conjecture the existence of functions able to discriminate non-isomorphic pairs for every instance of the problem.

Multivariate matching polynomials of cyclically labelled graphs

Discrete Mathematics, 2009

We consider the matching polynomials of graphs whose edges have been cyclically labelled with the ordered set of t labels {x 1 ,. . ., x t }. We first work with the cyclically labelled path, with first edge label x i , followed by N full cycles of labels {x 1 ,. . ., x t }, and last edge label x j. Let Φ i,N t+j denote the matching polynomial of this path. It satisfies the (τ, ∆)-recurrence: Φ i,N t+j = τ Φ i,(N −1)t+j −∆ Φ i,(N −2)t+j , where τ is the sum of all non-consecutive cyclic monomials in the variables {x 1 ,. . ., x t } and ∆ = (−1) t x 1 • • • x t. A combinatorial/algebraic proof and a matrix proof of this fact are given. Let G N denote the first fundamental solution to the (τ, ∆)-recurrence. We express G N (i) as a cyclic binomial using the Symmetric Representation of a matrix, (ii) in terms of Chebyshev polynomials of the second kind in the variables τ and ∆, and (iii) as a quotient of two matching polynomials. We extend our results from paths to cycles and rooted trees.

Matching polynomials: A matrix approach and its applications

Journal of the Franklin Institute, 1986

A new approach is formuiatedfor the matching polynomial m(G) of a graph G. A matrix A(G) is associated with G. A certain function de$ned on A(G) yields the matching polynomial of G. This approach leads to a simple characterization of m(G). It also facilitates a technique for constructing graphs with a given matching polynomial.

ON THE VERTEX DEGREE POLYNOMIAL OF GRAPHS

A novel graph polynomial, termed as vertex degree polynomial, has been conceptualized, and its discriminating power has been investigated regarding its coefficients and the coefficients of its derivatives and their relations with the physical and chemical properties of molecules. Correlation coefficients ranging from 95% to 98% were obtained using the coefficients of the first and second derivatives of this new polynomial. We also show the relations between this new graph polynomial, and two oldest Zagreb indices, namely the first and second Zagreb indices. We calculate the vertex degree polynomial along with its roots for some important families of graphs like tadpole graph, windmill graph, firefly graph, Sierpinski sieve graph and Kragujevac trees. Finally, we use the vertex degree polynomial to calculate the first and second Zagreb indices for the Dyck-56 network and also for the chemical compound triangular benzenoid G[r].

A characteristic polynomial for rooted graphs and rooted digraphs

Discrete Mathematics, 2001

We consider the one-variable characteristic polynomial p(G;) in two settings. When G is a rooted digraph, we show that this polynomial essentially counts the number of sinks in G. When G is a rooted graph, we give combinatorial interpretations of several coe cients and the degree of p(G;). In particular, |p(G; 0)| is the number of acyclic orientations of G, while the degree of p(G;) gives the size of the minimum tree cover (every edge of G is adjacent to some edge of T), and the leading coe cient gives the number of such covers. Finally, we consider the class of rooted fans in detail; here p(G;) shows cyclotomic behavior.

On the characteristic polynomial of the A_\alpha -matrix for some operations of graphs

Computational & Applied Mathematics, 2023

Let G be a graph of order n with adjacency matrix A(G) and diagonal matrix of degree D(G). For every α ∈ [0, 1], Nikiforov [1] defined the matrix A α (G) = αD(G) + (1 − α)A(G). In this paper we present the A α (G)-characteristic polynomial when G is obtained by coalescing two graphs, and if G is a semi-regular bipartite graph we obtain the A α-characteristic polynomial of the line graph associated to G. Moreover, if G is a regular graph we exhibit the A α-characteristic polynomial for the graphs obtained from some operations. Keywords A α-characteristic polynomial and Graph Operations and Eigenvalue.

The Characteristic Polynomials of Symmetric Graphs

Symmetry, 2018

In this paper, we study the way the symmetries of a given graph are reflected in its characteristic polynomials. Our aim is not only to find obstructions for graph symmetries in terms of its polynomials but also to measure how faithful these algebraic invariants are with respect to symmetry. Let p be an odd prime and Γ be a finite graph whose automorphism group contains an element h of order p. Assume that the finite cyclic group generated by h acts semi-freely on the set of vertices of Γ with fixed set F. We prove that the characteristic polynomial of Γ, with coefficients in the finite field of p elements, is the product of the characteristic polynomial of the induced subgraph Γ[F] by one of Γ \ F. A similar congruence holds for the characteristic polynomial of the Laplacian matrix of Γ.

USE OF EIGENVECTOR CENTRALITY TO DETECT GRAPH ISOMORPHISM

Graph Isomorphism is one of the classical problems of graph theory for which no deterministic polynomial-time algorithm is currently known, but has been neither proven to be NP-complete. Several heuristic algorithms have been proposed to determine whether or not two graphs are isomorphic (i.e., structurally the same). In this research, we propose to use the sequence (either the non-decreasing or nonincreasing order) of eigenvector centrality (EVC) values of the vertices of two graphs as a precursor step to decide whether or not to further conduct tests for graph isomorphism. The eigenvector centrality of a vertex in a graph is a measure of the degree of the vertex as well as the degrees of its neighbors. We hypothesize that if the non-increasing (or non-decreasing) order of listings of the EVC values of the vertices of two test graphs are not the same, then the two graphs are not isomorphic. If two test graphs have an identical non-increasing order of the EVC sequence, then they are declared to be potentially isomorphic and confirmed through additional heuristics. We test our hypothesis on random graphs (generated according to the Erdos-Renyi model) and we observe the hypothesis to be indeed true: graph pairs that have the same sequence of non-increasing order of EVC values have been confirmed to be isomorphic using the well-known Nauty software.