The characteristic polynomial and the matchings polynomial of a weighted oriented graph

Let $G^{\sigma }$ be a weighted oriented graph with skew adjacency matrix $S(G^{\sigma })$. Then $G^{\sigma }$ is usually referred as the weighted oriented graph associated to $S(G^{\sigma })$. Denote by $?(G^{\sigma }\text{;}\lambda )$ the characteristic polynomial of the weighted oriented graph $G^{\sigma }$, which is defined as$?(G^{\sigma }\text{;}\lambda )=\mathit{det}(\lambda I_n-S(G^{\sigma }))=\sum _{i=0}^n{a}_i(G^{\sigma }){\lambda }^{n-i}\text{.}$In this paper, we begin by interpreting all the coefficients of the characteristic polynomial of an arbitrary real skew symmetric matrix in terms of its associated oriented weighted graph. Then we establish recurrences for the characteristic polynomial and deduce a formula on the matchings polynomial of an arbitrary weighted graph. In addition, some miscellaneous results concerning the number of perfect matchings and the determinant of the skew adjacency matrix of an unweighted oriented graph are given.     

The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

The interlace polynomial of a graph

Motivated by circle graphs, and the enumeration of Euler circuits, we define a one-variable “interlace polynomial” for any graph. The polynomial satisfies a beautiful and unexpected reduction relation, quite different from the cut and fuse reduction characterizing the Tutte polynomial.It emerges that the interlace graph polynomial may be viewed as a special case of the Martin polynomial of an isotropic system, which underlies its connections with the circuit partition polynomial and the Kauffman brackets of a link diagram. The graph polynomial, in addition to being perhaps more broadly accessible than the Martin polynomial for isotropic systems, also has a two-variable generalization that is unknown for the Martin polynomial. We consider extremal properties of the interlace polynomial, its values for various special graphs, and evaluations which relate to basic graph properties such as the component and independence numbers.     

The minimal polynomial of a matrix

Let Rbe a finite dimensional central simple algebra over a field FA be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A(λ) of A over F. By using q_A(λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

Large deviation on a covering graph with group of polynomial growth

We discuss a large deviation principle of a periodic random walk on a covering graph with its transformation group of polynomial volume growth in view of geometry. As we shall observe, the behavior of a random walk at infinity is closely related to the Gromov?CHausdorff limit of an infinite graph and in the case where the graph admits an action of a group of polynomial volume growth, the Carnot-Carathéodory metric shows up in its limit space.     

The bivariate Ising polynomial of a graph

In this paper we discuss the two variable Ising polynomials in a graph theoretical setting. This polynomial has its origin in physics as the partition function of the Ising model with an external field. We prove some basic properties of the Ising polynomial and demonstrate that it encodes a large amount of combinatorial information about a graph. We also give examples which prove that certain properties, such as the chromatic number, are not determined by the Ising polynomial. Finally we prove that there exist large families of non-isomorphic planar triangulations with identical Ising polynomial.     

The minimal generating space of a polynomial

Given a polynomial $\text{P(X}{}_1\text{,…,X}{}_{\mathrm{N}}\text{)∈}\text{R}\text{[}\text{X}\text{]}$, we calculate a subspace G_p of the linear space 〈X〉 generated by the indeterminates which is minimal with respect to the property $\text{P∈}\text{R}\text{[}\text{G}{}_{\mathrm{p}}\text{]}$ (the algebra generated by G_p, and prove its uniqueness. Furthermore, we use this result to characterize the pairs (P,Q) of polynomials P(X₁,…,X_n) and Q(X₁,…,X_n) for which there exists an isomorphism T:〈X〉 →〈X〉 that “separates P from Q,” i.e., such that for some k(1<k<n) we can write P and Q as $\text{P}{}^1\text{(Y}{}_1\text{,…,Y}{}_{\mathrm{k}}\text{)}$ and $\text{Q}{}^1\text{(Y}{}_{\mathrm{<mtext>k+</mtext><mtext>1</mtext>}}\text{,…,Y}{}_{\mathrm{n}}\text{)}$ respectively, where $\text{Y}\text{=T}\text{X}$.     

The odd-valued chromatic polynomial of a signed graph

For a signed graph $\mathrm{\Sigma }=(G,\sigma )$, Zaslavsky defined a proper coloring on Σ and showed that the function counting the number of such colorings is a quasi-polynomial with period two, that is, a pair of polynomials, one for odd values and the other for even values. In this paper, we focus on the case of odd, written as $\chi (\mathrm{\Sigma },x)$ for short. We initially give a homomorphism expression of such colorings, by which the symmetry is considered in counting the number of homomorphisms. Besides, the explicit formulas $\chi (\mathrm{\Sigma },x)$ for some basic classes of signed graphs are presented. As a main result, we give a combinatorial interpretation of the coefficients in $\chi (\mathrm{\Sigma },x)$ and present several applications. In particular, the constant term in $\chi (\mathrm{\Sigma },x)$ gives a new criterion for balancing and a characterization for unbalanced unicyclic graph. At last, we also give a tight bound for the constant term of $\chi (\mathrm{\Sigma },x)$.     

On the chromatic polynomial of a graph

Let P(G,λ) be the chromatic polynomial of a graph G with n vertices, independence number α and clique number ω. We show that for every λ≥n, ()^α≤≤ () ^{n −ω}. We characterize the graphs that yield the lower bound or the upper bound.?These results give new bounds on the mean colour number μ(G) of G: n− (n−ω)() ^{n −ω}≤μ(G)≤n−α() ^α. Received: December 12, 2000 / Accepted: October 18, 2001?Published online February 14, 2002     

On the minimal polynomial of a resolvent

We derive the connection between the minimal polynomial of an operator and the minimal polynomial of its resolvent. An analogous result is obtained for the minimal polynomials of the iterated resolvents.     

The minimal cost maximum matching of a graph

Summary This paper deals with the problem of finding the maximum matching of a weighted graph, having the minimum cost. This problem is solved via a branch and bound algorithm, derived directly fromLand andDoig's technique. The linear programming problem associated with each step of the procedure is solved through a primal-dual algorithm.

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Zusammenfassung Diese Arbeit beschäftigt sich mit demmatching problem. In einem gewichteten Graphen ist dasmaximal matching mit minimalen Kosten gesucht. Zur Lösung wird ein vonLand undDoig beschriebener Branch-and-Bound-Algorithmus verwendet. Das sich je Iterationsschritt ergebende LP-Problem wird mit einem Primal-Dual-Algorithmus gelöst.

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The authors are with the Laboratory of Automatic Control, Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano, I-20133 Milano, Piazza Leonardo da Vinci, 32.     

GMRES and the minimal polynomial

We present a qualitative model for the convergence behaviour of the Generalised Minimal Residual (GMRES) method for solving nonsingular systems of linear equationsAx =b in finite and infinite dimensional spaces. One application of our methods is the solution of discretised infinite dimensional problems, such as integral equations, where the constants in the asymptotic bounds are independent of the mesh size.Our model provides simple, general bounds that explain the convergence of GMRES as follows: If the eigenvalues ofA consist of a single cluster plus outliers then the convergence factor is bounded by the cluster radius, while the asymptotic error constant reflects the non-normality ofA and the distance of the outliers from the cluster. If the eigenvalues ofA consist of several close clusters, then GMRES treats the clusters as a single big cluster, and the convergence factor is the radius of this big cluster. We exhibit matrices for which these bounds are tight.Our bounds also lead to a simpler proof of existing r-superlinear convergence results in Hilbert space.This research was partially supported by National Science Foundation grants DMS-9122745, DMS-9423705, CCR-9102853, CCR-9400921, DMS-9321938, DMS-9020915, and DMS-9403224.     

Vectors with a prescribed minimal polynomial

Let V be a finite dimensional vector space over the field Fand φ (x)∈F[x].LetxV → V be a linear operator. Let S_φbe the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S _φ to be a subspace.     

Vectors with a prescribed minimal polynomial

Let V be a finite dimensional vector space over the field Fand φ (x)?F[x].Letx V → V be a linear operator. Let S_φ be the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S _φ to be a subspace.     

The chromatic polynomial of an unlabeled graph

We investigate the chromatic polynomial χ(G, λ) of an unlabeled graph G. It is shown that $\text{χ(G, λ) = (}\text{1}\text{|A(g)|}\text{) Σ}{}_{\mathrm{\pi \; \in \; A(g)}}\text{χ(g, π, λ)}$, where g is any labeled version of G, A(g) is the automorphism group of g and χ(g, π, λ) is the chromatic polynomial for colorings of g fixed by π. The above expression shows that χ(G, λ) is a rational polynomial of degree n = |V(G)| with leading coefficient $\text{1}\text{|A(g)|}$. Though χ(G, λ) does not satisfy chromatic reduction, each polynomial χ(g, π, λ) does, thus yielding a simple method for computing χ(G, λ). We also show that the number N(G) of acyclic orientations of G is related to the argument λ = ?1 by the formula $\text{N(G) = (}\text{1}\text{|A(g)|}\text{) Σ}{}_{\mathrm{\pi \; \in \; A(g)}}\text{(?1)}{}^{\mathrm{s(\pi )}}\text{χ(g, π, ?1)}$, where s(π) is the number of cycles of π. This information is used to derive Robinson's (“Combinatorial Mathematics V” (Proc. 5th Austral. Conf. 1976), Lecture Notes in Math. Vol. 622, pp. 28–43, Springer-Verlag, New York/Berlin, 1977) cycle index sum equations for counting unlabeled acyclic digraphs.     

On conjugate characteristic polynomial of a graph

Let $c=a+b\sqrt{m}$ and $\overline{c}=a-b\sqrt{m}$ , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A ^c =[c _ij ] is the conjugate adjacency matrix of a graph G if c _ij =c for any two adjacent vertices i and j, $c_{ij}=\overline{c}$ for any two nonadjacent vertices i and j, and c _ij =0 if i=j. Let P _G ^c (λ)=|λ I?A ^c | denote the conjugate characteristic polynomial of G. Further, let e=e(G) and Δ=Δ(G) be the number of edges and number of triangles of G, respectively. Let G and H be two graphs of order n and let e(G)=e(H). In this work we prove that c ₃(G)=c ₃(H) if and only if Δ(G)=Δ(H) and $\Delta(\overline{G})=\Delta(\overline{H})$ , where $\overline{G}$ denotes the complement of G and c _k is the coefficient which corresponds to λ ^n?k with respect to P _G ^c (λ). Besides, we here give the conjugate spectrum and conjugate characteristic polynomial of all connected graphs of order n=2,3,4,5, with respect to the constant $c=1+\sqrt{2}$ .     

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.