Algebra Of Graphs Research Articles

Eight Times Four Bialgebras of Hypergraphs, Cointeractions, and Chromatic Polynomials

Abstract The bialgebra of hypergraphs, a generalization of W. Schmitt’s Hopf algebra of graphs [25], is shown to have a cointeracting bialgebra structure, giving a double bialgebra in the sense of L. Foissy, who has recently proven [15] that there is then a unique double bialgebra morphism to the double bialgebra structure on the polynomial ring ${\mathbb Q}[x]$. We show that the associated polynomial is the hypergraph chromatic polynomial. Moreover, hypergraphs occur in quartets: There is a dual, a complement, and a dual complement hypergraph. These correspondences are involutions and give rise to three other double bialgebras, and three more chromatic polynomials. In addition to these two quartets of bialgebras we give six more, including recent bialgebras of hypergraphs introduced by M. Aguiar and F. Ardila [1] and by L. Foissy [17].

English

In the formulation of his celebrated Formality conjecture, M. Kontsevich introduced a universal version of the deformation theory for the Schouten algebra of polyvector fields on affine manifolds. This universal deformation complex takes the form of a differential graded Lie algebra of graphs, denoted $\mathsf{fGC}_2$, together with an injective morphism towards the Chevalley-Eilenberg complex associated with the Schouten algebra. The latter morphism is given by explicit local formulas making implicit use of the supergeometric interpretation of the Schouten algebra as the algebra of functions on a graded symplectic manifold of degree $1$. The ambition of the present work is to generalise Kontsevich's construction to graded symplectic manifolds of arbitrary degree $n\geq1$. The corresponding graph model is given by the full Kontsevich graph complex $\mathsf{fGC}_d$ where $d=n+1$ stands for the dimension of the associated AKSZ type $\sigma$-model. This generalisation is instrumental to classify universal structures on graded symplectic manifolds. In particular, the zeroth cohomology of the full graph complex $\mathsf{fGC}_{d}$ is shown to act via $\mathsf{Lie}_\infty$-automorphisms on the algebra of functions on graded symplectic manifolds of degree $n$. This generalises the known action of the Grothendieck-Teichm\"{u}ller algebra $\mathfrak{grt}_1\simeq H^0(\mathsf{fGC}_2)$ on the space of polyvector fields. This extended action can in turn be used to generate new universal deformations of Hamiltonian functions, generalising Kontsevich flows on the space of Poisson manifolds to differential graded manifolds of higher degrees. As an application of the general formalism, universal deformations of Courant algebroids via trivalent graphs are presented.

Algebra of shares, complete bipartite graphs and $\mathfrak{sl}_2$ weight system

A function of chord diagrams is called a weight system if it satisfies the so-called four-term relations. Vassiliev's theory describes finite-order knot invariants in terms of weight systems. In particular, there is a weight system corresponding to the coloured Jones polynomial. This weight system is described in terms of the Lie algebra $\mathfrak{sl}_2$. According to the Chmutov-Lando theorem, the value of this weight system depends only on the intersection graph of the chord diagram. Therefore, it is possible to discuss the values of this weight system at intersection graphs. We obtain formulae for the generating functions of the values of the $\mathfrak{sl}_2$ weight system at complete bipartite graphs. Using these formulae we prove that Lando's conjecture about the degree of the polynomial that is the value of this weight system at the projection of a graph onto the subspace of primitive elements in the Hopf algebra of graphs is true for complete bipartite graphs and for a certain wider class of graphs. We introduce the algebra of shares and the $\mathfrak{sl}_2$ weight system on shares. These are the main tools for our proof. Bibliography: 14 titles.

The Terwilliger algebra of symplectic dual polar graphs, the subspace lattices and Uq(sl2)

The adjacency matrix of a symplectic dual polar graph restricted to the eigenspaces of an abelian automorphism subgroup is shown to act as the adjacency matrix of a weighted subspace lattice. The connection between the latter and Uq(sl2) is used to find the irreducible components of the standard module of the Terwilliger algebra of symplectic dual polar graphs. The multiplicities of the isomorphic submodules are given.

On the Hopf algebra of multi-complexes

We introduce a general class of combinatorial objects, which we call multi-complexes, which simultaneously generalizes graphs, multigraphs, hypergraphs and simplicial and delta complexes. We introduce a natural algebra of multi-complexes which is defined as the algebra which has a formal basis $${\mathscr {C}}$$ of all isomorphism types of multi-complexes, and multiplication is to take the disjoint union. This is a Hopf algebra with an operation encoding the disassembly information for such objects and extends the Hopf algebra of graphs. In our main result, we explicitly describe here the structure of this Hopf algebra of multi-complexes H. We find an explicit basis $${\mathscr {B}}$$ of the space of primitives, which is of combinatorial relevance: it is such that each multi-complex is a polynomial with non-negative integer coefficients of the elements of $${\mathscr {B}}$$ , and each $$b\in {\mathscr {B}}$$ is a polynomial with integer coefficients in $${\mathscr {C}}$$ . Using this, we find the grouping free formula for the antipode. The coefficients appearing in all these polynomials are, up to signs, numbers counting multiplicities of sub-multi-complexes in a multi-complex. We also explicitly illustrate how our results specialize to the graph Hopf algebra, and observe how they specialize to results in all of the above-mentioned particular cases. We also investigate applications of these results to the graph reconstruction conjectures and rederive some results in the literature on these questions.

Values of the weight system on a family of graphs that are not the intersection graphs of chord diagrams

The Chmutov-Lando theorem claims that the value of a weight system (a function on the chord diagrams that satisfies the four-term Vassiliev relations) corresponding to the Lie algebra depends only on the intersection graph of the chord diagram. We compute the values of the weight system at the graphs in several infinite series, which are the joins of a graph with a small number of vertices and a discrete graph. In particular, we calculate these values for a series in which the initial graph is the cycle on five vertices; the graphs in this series, apart from the initial one, are not intersection graphs. We also derive a formula for the generating functions of the projections of graphs equal to the joins of an arbitrary graph and a discrete graph to the subspace of primitive elements of the Hopf algebra of graphs. Using the formula thus obtained, we calculate the values of the weight system at projections of the graphs of the indicated form onto the subspace of primitive elements. Our calculations confirm Lando’s conjecture concerning the values of the weight system at projections onto the subspace of primitives. Bibliography: 17 titles.

The Hopf algebra structure of the R\u2217-operation

We give a Hopf-algebraic formulation of the R∗-operation, which is a canonical way to render UV and IR divergent Euclidean Feynman diagrams finite. Our analysis uncovers a close connection to Brown’s Hopf algebra of motic graphs. Using this connection we are able to provide a verbose proof of the long observed ‘commutativity’ of UV and IR subtractions. We also give a new duality between UV and IR counterterms, which, entirely algebraic in nature, is formulated as an inverse relation on the group of characters of the Hopf algebra of log-divergent scaleless Feynman graphs. Many explicit examples of calculations with applications to infrared rearrangement are given.

Values of the $$\mathfrak{sl}_2$$ Weight System on Complete Bipartite Graphs

A weight system is a function on chord diagrams that satisfies the so-called four-term relations. Vassiliev's theory of finite-order knot invariants describes these invariants in terms of weight systems. In particular, there is a weight system corresponding to the colored Jones polynomial. This weight system can be easily defined in terms of the Lie algebra $\mathfrak{sl}_2$, but this definition is too cumbersome from the computational point of view, so that the values of this weight system are known only for some limited classes of chord diagrams. In the present paper we give a formula for the values of the $\mathfrak{sl}_2$ weight system for a class of chord diagrams whose intersection graphs are complete bipartite graphs with no more than three vertices in one of the parts. Our main computational tool is the Chmutov--Varchenko reccurence relation. Furthermore, complete bipartite graphs with no more than three vertices in one of the parts generate Hopf subalgebras of the Hopf algebra of graphs, and we deduce formulas for the projection onto the subspace of primitive elements along the subspace of decomposable elements in these subalgebras. We compute the values of the $\mathfrak{sl}_2$ weight system for the projections of chord diagrams with such intersection graphs. Our results confirm certain conjectures due to S.K.Lando on the values of the weight system $\mathfrak{sl}_2$ at the projections of chord diagrams on the space of primitive elements.

On the Terwilliger algebra of distance-biregular graphs

Let Γ denote a distance-biregular graph with vertex set X. Fix x∈X and let T=T(x) denote the Terwilliger algebra of Γ with respect to x. In this paper we consider irreducible T-modules with endpoint 1. We show that there are no such modules if and only if Γ is the complete bipartite graph K1,n(n≥1) and x is a vertex of Γ with valency 1. If the valency of x is at least 2 then we show that up to isomorphism there is a unique irreducible T-module of endpoint 1, and this module is thin.

Invariants of Framed Graphs and the Kadomtsev—Petviashvili Hierarchy

S. V. Chmutov, M. E. Kazarian, and S. K. Lando have recently introduced a class of graph invariants, which they called shadow invariants (these invariants are graded homomorphisms from the Hopf algebra of graphs to the Hopf algebra of polynomials in infinitely many variables). They proved that, after an appropriate rescaling of the variables, the result of the averaging of almost every such invariant over all graphs turns into a linear combination of single-part Schur functions and, thereby, becomes a τ-function of an integrable Kadomtsev-Petviashvili hierarchy. We prove a similar assertion for the Hopf algebra of framed graphs. At the same time, we show that there is no such an analogue for a number of other Hopf algebras of a similar nature, in particular, for the Hopf algebras of weighted graphs, chord diagrams, and binary delta-matroids. Thus, it turns out that the Hopf algebras of graphs and framed graphs are distinguished among the graded Hopf algebras of combinatorial nature.

PRIMITIVE IDEALS AND PURE INFINITENESS OF ULTRAGRAPH C ⁎ -ALGEBRAS

Let $\mathcal{G}$ be an ultragraph and let $C^*(\mathcal{G})$ be the associated $C^*$-algebra introduced by Mark Tomforde. For any gauge invariant ideal $I_{(H,B)}$ of $C^*(\mathcal{G})$, we approach the quotient $C^*$-algebra $C^*(\mathcal{G})/I_{(H,B)}$ by the $C^*$-algebra of finite graphs and prove versions of gauge invariant and Cuntz-Krieger uniqueness theorems for it. We then describe primitive gauge invariant ideals and determine purely infinite ultragraph $C^*$-algebras (in the sense of Kirchberg-R$o$rdam) via Fell bundles.

Differential algebra of cubic planar graphs

In this article we associate a combinatorial differential graded algebra to a cubic planar graph G. This algebra is defined combinatorially by counting binary sequences, which we introduce, and several explicit computations are provided. In addition, in the appendix by K. Sackel the Fq-rational points of its graded augmentation variety are shown to coincide with (q+1)-colorings of the dual graph.

On linear algebra of balance-binomial graphs

In this work, we introduce balance-binomial graphs, whose entries of adjacency matrix are balance-binomial coefficients. Then we obtain some characteristic properties of the graph.

Algebraic graphs with class (functional pearl)

The paper presents a minimalistic and elegant approach to working with graphs in Haskell. It is built on a rigorous mathematical foundation --- an algebra of graphs --- that allows us to apply equational reasoning for proving the correctness of graph transformation algorithms. Algebraic graphs let us avoid partial functions typically caused by `malformed graphs' that contain an edge referring to a non-existent vertex. This helps to liberate APIs of existing graph libraries from partial functions. The algebra of graphs can represent directed, undirected, reflexive and transitive graphs, as well as hypergraphs, by appropriately choosing the set of underlying axioms. The flexibility of the approach is demonstrated by developing a library for constructing and transforming polymorphic graphs.

Rational mixed Tate motivic graphs

In this paper, we study the combinatorics of a subcomplex of the Bloch-Kriz cycle complex [4] used to construct the category of mixed Tate motives. The algebraic cycles we consider properly contain the subalgebra of cycles that correspond to multiple logarithms (as defined in [12]). We associate an algebra of graphs to our subalgebra of algebraic cycles. We give a purely graphical criterion for admissibilty. We show that sums of bivalent graphs correspond to coboundary elements of the algebraic cycle complex. Finally, we compute the Hodge realization for an infinite family of algebraic cycles represented by sums of graphs that are not describable in the combinatorial language of [12].

Antipodes and involutions

If H is a connected, graded Hopf algebra, then Takeuchi's formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmetric functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidence Hopf algebra of graphs. We also find cancellation-free expressions for particular values of the antipode in the immaculate basis for the noncommutative symmetric functions as well as the Malvenuto–Reutenauer and Poirier–Reutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research.

On the Terwilliger algebra of bipartite distance-regular graphs with Δ2 = 0 and c2 = 1

Let Γ denote a bipartite distance-regular graph with diameter D≥4 and valency k≥3. Let X denote the vertex set of Γ, and let A denote the adjacency matrix of Γ. For x∈X and for 0≤i≤D, let Γi(x) denote the set of vertices in X that are distance i from vertex x. Define a parameter Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. We first show that Δ2=0 implies that D≤5 or c2∈{1,2}.For x∈X let T=T(x) denote the subalgebra of MatX(C) generated by A,E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. By the endpoint of an irreducible T-module W we mean min{i|Ei⁎W≠0}.In this paper we assume Γ has the property that for 2≤i≤D−1, there exist complex scalars αi, βi such that for all x,y,z∈X with ∂(x,y)=2, ∂(x,z)=i, ∂(y,z)=i, we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|. We additionally assume that Δ2=0 with c2=1.Under the above assumptions we study the algebra T. We show that if Γ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible T-module with endpoint 2. We give an orthogonal basis for this T-module, and we give the action of A on this basis.

Towards Compositional Graph Theory

Decomposing graphs into simpler graphs is one of the central concerns of graph theory. Investigations have revealed deep concepts such as modular decomposition, tree width or rank width, which measure—in different ways—the structural complexity of a graph's topology. Courcelle and others have shown that such concepts can be used to obtain efficient algorithms for families of graphs that are amenable to decomposition (e.g. those that have bounded tree-width). These algorithms, in turn, are of course of use in computer science, where graphs are ubiquitous. In this paper we take the first steps towards understanding notions of decomposition in graph theory compositionally, and more generally, in a categorical setting: category theory, after all, is the mathematics of compositionality.We introduce the concept of ∪–matrices (cup-matrices). Like ordinary matrices, ∪-matrices are the arrows of a PROP: we give a presentation, extending the work of Lafont, and Bonchi, Zanasi and the second author. A variant of ∪-matrices is then used in the development of a novel algebra of simple graphs, the lingua franca of graph theory. The algebra is that of a certain symmetric monoidal theory: ∪-matrices—akin to adjacency matrices—encode the graphs' topology.

The Terwilliger algebra of the incidence graphs of Johnson geometry, II

Let J(n,m,m+1) denote the incidence graph of Johnson geometry. It is well known that the Johnson graph J(n,m) is a halved graph of J(n,m,m+1). Let T and T′ be the Terwilliger algebra of J(n,m,m+1) and J(n,m), respectively. In this paper, we focus on the structures of irreducible T-modules, and then completely determine T. Furthermore, we discover the relationship between irreducible modules of T and T′. As a result, the algebra T′ is determined again.

Methods of Automatic Control of the Topological State of the Equipment of Electrical Substations

Topological states of substations’ equipment are determined by the states of switching devices. The considered classes of topological states are the states of: individual devices, groups of electrical devices, and substations as a whole. Along with independent significance of these characteristics their role as logical locks when switching is considered. Topological characteristics are calculated by analyzing the topology of electrical circuits and substations. For this, the mathematical formalism has been developed, which is based on the special algebra of graphs. Formulas are given to calculate the topological states of the equipment of substation.