A Hopf algebra on subgraphs of a graph

TL;DR

This paper constructs a Hopf algebra structure on subgraphs of a graph, explores its dual, and studies algebra morphisms induced by graph homomorphisms, establishing a functor between graph and algebra categories.

Contribution

It introduces a novel Hopf algebra framework on subgraphs and analyzes its dual and functorial properties, linking graph theory and algebra.

Findings

Established a Hopf algebra on subgraphs

Derived the dual Hopf algebra structure

Created a covariant functor from graph to algebra categories

Abstract

In this paper, we construct a bialgebraic and further a Hopf algebraic structure on top of subgraphs of a given graph. Further, we give the dual structure of this Hopf algebraic structure. We study the algebra morphisms induced by graph homomorphisms, and obtain a covariant functor from a graph category to an algebra category.

Equations

$$a\cdot (b\otimes c):=ab\otimes c\text{ and }(b\otimes c)\cdot a:=b\otimes ca.$$

$$(x\cdot y)\cdot z=x\cdot (y\cdot z)\text{ for all }x,y,z\in S.$$

$$x\cdot 1=1\cdot x=x\text{ for all }x\in S.$$

$$\varphi (x{\cdot }_{S}y)=\varphi (x){\cdot }_T\varphi (y)\text{ and }\varphi (1_S)=1_T\text{ for all }x,y\in S.$$

$$f(x+y)=f(x)+f(y)\text{ and }f(kx)=kf(x)\text{ for all }x,y\in M,k\in \mathbf{k}.$$

$$F={\Gamma }_1\cdots {\Gamma }_n,\text{ where }n\ge 0\text{ and }{\Gamma }_1,\cdots ,{\Gamma }_n\in \mathcal{G}.$$

$${\rm bre}(\mathbbm{1})=0,\,{\rm bre}(\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture})=1,\,{\rm bre}(\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture})=4.$$

$$H(\mathcal{G}):=\mathbf{k}M(\mathcal{G})$$

$$\displaystyle{\mathcal{G}}=\{\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture},\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture},\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\},\,M({\mathcal{G}})=\langle\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture},\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture},\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\rangle\,\text{ and }\,H({\mathcal{G}})={\bf k}\langle\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture},\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture},\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\rangle.$$

$$\mathcal{G}=$$

$$M(\mathcal{G})=$$

$$H(\mathcal{G})=$$

$$C\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbf{k}\otimes C$$

$${\beta }_l:C\to \mathbf{k}\otimes C,x\mapsto 1_{\mathbf{k}}\otimes x,$$

$${\beta }_r:C\to C\otimes \mathbf{k},x\mapsto x\otimes 1_{\mathbf{k}}\text{ for }x\in C.$$

$$\tau :C\otimes C\to C\otimes C,x\otimes y\to y\otimes x\text{ for }x,y\in C,$$

$${\Delta }_G(F):={\Delta }_G(\text{mathbbm}1):=\text{mathbbm}1\otimes \text{mathbbm}1.$$

$${\Delta }_G(F):=\sum _{V_1\bigsqcup V_2=V(F)}F[V_1]\otimes F[V_2],$$

$$F[V_1],F[V_2]\in M(\mathcal{G})\text{ and }{\Delta }_G(F)\in H(\mathcal{G})\otimes H(\mathcal{G}).$$

$${\Delta }_G(F):={\Delta }_G({\Gamma }_1\cdots {\Gamma }_n):={\Delta }_G({\Gamma }_1)\cdots {\Delta }_G({\Gamma }_n).$$

$${\Delta }_G(F)\in H(\mathcal{G})\otimes H(\mathcal{G})\text{ by }{\Delta }_G({\Gamma }_i)\in H(\mathcal{G})\otimes H(\mathcal{G})\text{ for each }i=1,\cdots ,n.$$

$$\displaystyle\Delta_{G}\left(\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\right)=$$

$$\displaystyle\Delta_{G}\left(\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\right)=$$

$$=$$

$$\displaystyle+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}$$

$$\displaystyle+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}$$

$$\displaystyle+\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\otimes\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}+\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}+\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 2} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture}+\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny$\tiny 1$} \put(6.0,3.0){\tiny$\tiny 2$} \end{picture}\otimes 1.$$

$${\Delta }_G(F):=\sum _{U\uplus V=V(F)}F[U]\otimes F[V],$$

$$U=U_1\cup \cdots \cup U_n\text{ and }V=V_1\cup \cdots \cup V_n,$$

$$U_i\bigsqcup V_i=V({\Gamma }_i)\text{ for }i=1,\cdots ,n.$$

$$V({\Gamma }_1)=\{1,2\}\uplus \varnothing =\{1\}\uplus \{2\}=\{2\}\uplus \{1\}=\varnothing \uplus \{1,2\},$$

$$V({\Gamma }_2)=\{3,4\}\uplus \varnothing =\{3\}\uplus \{4\}=\{4\}\uplus \{3\}=\varnothing \uplus \{3,4\},$$

$$\Delta_{G}(\Gamma_{1})=\Delta_{G}(\,\,\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny$1$} \put(11.0,-2.0){\tiny$2$} \end{picture}\,\,)=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny$1$} \put(11.0,-2.0){\tiny$2$} \end{picture}\,\,\otimes\mathbbm{1}+\mathbbm{1}\otimes\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny$1$} \put(11.0,-2.0){\tiny$2$} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny$1$} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny$2$} \end{picture}+\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny$2$} \end{picture}\otimes\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny$1$} \end{picture},$$

Full text

Keywords: Hopf algebra, covariant functor, graph category, algebra category

1 Introduction

The Hopf algebraic theory has a long history and broad applications in mathematics and physics [2, 8, 14, 22, 24, 27, 28, 30, 33]. The Hopf algebra of rooted forests originated from the work of A. Dür [5] and had a very rich theory. For example, A. Connes and D. Kreimer introduced the Connes-Kreimer Hopf algebra of rooted forests in [2, 23] to study the renormalization of perturbative quantum field theory. In [2, 3, 4], they introduced commutative Hopf algebras of graphs to study the combinatorial structure of renormalization. It is also related to the Loday-Ronco Hopf algebra [26, 32] and Grossman-Larson Hopf algebra [13] of rooted trees. L. Foissy has done a lot of work on the Hopf algebra of rooted forests [8, 9, 10, 11]. As related results, various infinitesimal bialgebras on decorated rooted forests have been established in [16, 17, 31], via different Hochschild 1-cocycle conditions.

A totally assigned graph is a graph with a total order on its edges. In [6], G. E. Duchamp et al. built a Hopf algebra of totally assigned graphs. L. Foissy [12] constructed a Hopf algebra on all graphs to insert chromatic polynomial into the theory of combinatorial Hopf algebra, and gave a new proof of some classical results.

The concept of graph (incidence) Hopf algebras was introduced by Schmitt [29]. Namely, a graph algebra is a commutative, cocommutative, graded, connected Hopf algebra, whose basis elements correspond to finite graphs. Humpert and Martin [20] obtained a new nonrecursive formula for the antipode of a graph (incidence) Hopf algebra. In [25], the author introduced the $4$-bialgebra of graphs that satisfies some relations.

The study of Hopf algebras of graphs is extensive, but the research on Hopf algebras of subgraphs has no result yet. Subgraph plays an important role in the study of graph theory. For example, we can study the prefect graph by the independence number and clique number of the induced subgraph. We can determine if a graph is an interval map by the induced subgraph and many more.

In the present paper, we construct a Hopf algebraic structure on subgraphs of a given graph. In particular, a combinatorial description of the coproduct is given. Narrowing ourself to the algebraic part of the aforementioned Hopf algebra on subgraphs of a given graph, we obtain an algebra morphism from a graph homomorphism. Using the language of categories, we obtain a covariant functor from the category of graphs to the category of algebras.

Here is the structure of the paper. In Section 2, we first review the definitions of free monoid and free module. Then we proceed to give an algebraic structure on subgraphs of a graph. Next, we construct a coalgebraic structure on subgraphs of a graph (Lemma 2.13), thereby we construct a bialgebra on the subgraph (Theorem 2.17). Finally, to make the coproduct $\Delta_{G}$ more explicit, we describe it in a combinatorial way (Eq. (5)).

In Section 3, continuing the line in Section 2, a Hopf algebra structure on subgraphs of a graph is given (Theorem 3.5). Further, we study the dual Hopf algebra structure. We conclude with a description of the dual Hopf algebra (Theorem 3.7). This section is also devoted to an algebra morphism induced by a graph homomorphism (Theorem 3.12). As an application, we obtain a functor from graph category to algebra category (Theorem 3.15).

Notation. Graphs considered in this paper are connected and undirected graphs without multiple edges and loops. We will be working over a unitary commutative base ring ${\bf k}$. Linear maps and tensor products are taken over ${\bf k}$. For an algebra $A$, we view $A\otimes A$ as an $A$-bimodule via

[TABLE]

2 A bialgebra on subgraphs of a given graph

In this section, we build a bialgebraic structure on top of subgraphs of a given graph.

2.1 An algebraic structure on subgraphs of a given graph

Let us first review some notations on graphs and algebras, which are used throughout the remainder of the paper. For a graph $G$, denote by $V(G)$ and $E(G)$ the vertex set and edge set of $G$, respectively.

Definition 2.1**.**

[7] Let $G$ be a graph.

1. (a)

If its $n$ vertices are distinguished from one another by labels such as $\{1,2\cdots,n\}$, then $G$ is said to be labeled. 2. (b)

If there is a path between any two vertices of $G$, then $G$ is connected. 3. (c)

A subgraph of a graph $G$ is simply a graph, all of whose vertices belong to $V(G)$ and all of whose edges belong to $E(G)$. 4. (d)

For $S\subseteq V(G)$, the induced subgraph $G[S]$ in $G$ by $S$ is the graph whose edge set consists of all of the edges in $G$ that have both endpoints in $S$.

The concepts of free monoid and free module are needed later.

Definition 2.2**.**

[18]

1. (a)

A semigroup is a nonempty set $S$ together with a binary operation $\cdot:S\times S\to S$ which is associative:

[TABLE]

A semigroup $S$ is called a monoid if it contains an element $1$ with the property that

[TABLE] 2. (b)

Let $(S,\cdot_{S},1_{S})$ and $(T,\cdot_{T},1_{T})$ be monoids. A map $\phi:S\rightarrow T$ is called a monoid morphism if

[TABLE] 3. (c)

The **free monoid ** on a set $X$ is a monoid $S$ together with a map $i:X\rightarrow S$ with the property that, for any monoid $T$ and map $\phi:X\rightarrow T$, there exists a unique monoid morphism $\psi:S\rightarrow T$ such that $\psi\circ i=\phi$.

Definition 2.3**.**

[21]

1. (a)

Let ${\bf k}$ be a ring. A (left) ${\bf k}$-module is an additive abelian group $M$ together with a function ${\bf k}\times M\rightarrow M$, $(k,x)\mapsto kx$ such that for all $k,l\in{\bf k}$ and $x,y\in M$:

1. (i)

$k(x+y)=kx+ky;$ 2. (ii)

$(k+l)x=kx+lx;$ 3. (iii)

$k(lx)=(kl)x.$ 2. (b)

Let $M$ and $N$ be modules over ring ${\bf k}$. A map $f:M\rightarrow N$ is a ${\bf k}$-module morphism if

[TABLE] 3. (c)

The free ${\bf k}$-module on a set $X$ is a ${\bf k}$-module $F$ together with a set map $i:X\rightarrow F$ with the property that, for any ${\bf k}$-module $M$ and map $f:X\rightarrow M$, there exists a unique ${\bf k}$-module morphism $\bar{f}:F\rightarrow M$ such that $\bar{f}\circ i=f$.

A ${\bf k}$-module $M$ is free if and only if it has a ${\bf k}$-linear basis [21].

In the rest of this paper, let $G$ be a graph with vertices labelled by natural numbers and let ${\mathcal{G}}$ be the set of nonempty connected subgraphs of $G$. Let $M({\mathcal{G}})=\langle{\mathcal{G}}\rangle$ be the free monoid on ${\mathcal{G}}$ in which the multiplication is the concatenation, denoted by $m_{G}$ and usually suppressed. The unit in $M({\mathcal{G}})$ is the empty graph, denote by $\mathbbm{1}$. An element $F$ in $M({\mathcal{G}})$ is a noncommutative product of connected subgraphs in ${\mathcal{G}}$:

[TABLE]

Here we employ the convention that $F=\mathbbm{1}$ whenever $n=0$. We define ${\rm bre}(F):=n$ to be the breadth of $F$. For example,

[TABLE]

Denote by

[TABLE]

the free ${\bf k}$-module spanned by $M({\mathcal{G}})$. By linearity, the multiplication $m_{G}$ can be extended to $H({\mathcal{G}})$ and then the triple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1})$ is an algebra. Indeed, it is the free algebra on the set ${\mathcal{G}}$.

Example 2.4**.**

Here are some examples of $H({\mathcal{G}})$.

1. (a)

If $G=\bullet_{1}$, then ${\mathcal{G}}=\{\bullet_{1}\}$, $M({\mathcal{G}})=\langle\bullet_{1}\rangle$ and $H({\mathcal{G}})={\bf k}\langle\bullet_{1}\rangle$. 2. (b)

If G=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 1} \put(6.0,3.0){\tiny\tiny 2} \end{picture}, then

[TABLE] 3. (c)

If G=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(8.0,5.0){\circle*{2.0}} \put(3.0,5.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(8.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny2} \put(-2.0,4.0){\tiny1} \put(11.0,-2.0){\tiny3} \put(11.0,4.0){\tiny4} \end{picture}  , then

[TABLE]

2.2 A coproduct on subgraphs of a given graph

In this subsection, we construct a coproduct on subgraphs of a graph. Let us first recall

Definition 2.5**.**

[27, Definition 2.1] A coalgebra $(C,\,\Delta,\,\varepsilon)$ over ${\bf k}$ is a ${\bf k}$-module $C$ together with morphisms of ${\bf k}$-modules $\Delta:=C\rightarrow C\otimes C$, called the coproduct, and $\varepsilon:=C\rightarrow{\bf k}$, called the counit, such that the diagrams

[TABLE]

and

[TABLE]

are commutative. Here $\beta_{l}$ and $\beta_{r}$ are the isomorphisms defined by

[TABLE]

If a coalgebra $(C,\,\Delta,\,\varepsilon)$ satisfies $\tau\circ\Delta=\Delta$, where $\tau$ is a ${\bf k}$-linear map

[TABLE]

then the coalgebra $(C,\,\Delta,\,\varepsilon)$ is said to be a cocomutative coalgebra.

Definition 2.6**.**

[1, p. 54] Let $(C,\,\Delta,\,\varepsilon)$ be a coalgebra. A submodule $D$ of a coalgebra $C$ is called a ${\bf k}$-subcoalgebra of $C$ if $D$ is a coalgebra with the restriction of $\Delta$ and $\varepsilon$. Further, $D$ is called a simple ${\bf k}$-subcoalgebra if it does not have any ${\bf k}$-subcoalgebras other than $\{0\}$ and $D$.

Now we define a coproduct $\Delta_{G}$ on $H({\mathcal{G}})$. By linearity, we only need to define $\Delta_{G}(F)$ for basis elements $F\in M({\mathcal{G}})$. If ${\rm bre}(F)=0$, then $F=\mathbbm{1}$ and define

[TABLE]

If ${\rm bre}(F)=1$, then $F\in{\mathcal{G}}$ and we define

[TABLE]

where $F[V_{i}]$ are induced subgraphs of $F$ by vertex sets $V_{i}$ with $i=1,2$. Note

[TABLE]

If ${\rm bre}(F)>1$, then $F=\Gamma_{1}\cdots\Gamma_{n}$ for some $n\geq 2$ and $\Gamma_{1},\ldots,\Gamma_{n}\in{\mathcal{G}}$, and define

[TABLE]

Again note that

[TABLE]

Let us give an example for better insight into $\Delta_{G}$.

Example 2.7**.**

Let G=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 1} \put(6.0,3.0){\tiny\tiny 2} \end{picture}. Then \tiny 1$$\tiny 2 and \tiny 1$$\tiny 2$$\tiny 1$$\tiny 2 are in $M({\mathcal{G}})$, and

[TABLE]

2.3 A combinatorial description of the coproduct

This subsection is devoted to a combinatorial description of the coproduct $\Delta_{G}$ given in Subsection 2.2.

Let $G$ be a graph and $F=\Gamma_{1}\cdots\Gamma_{n}\in M({\mathcal{G}})$ with $\Gamma_{1},\cdots,\Gamma_{n}\in{\mathcal{G}}$. Note that $V(F)$ is a multiset. For example, let G=\,\,\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(8.0,5.0){\circle*{2.0}} \put(3.0,5.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(8.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny2} \put(-2.0,4.0){\tiny1} \put(11.0,-2.0){\tiny3} \put(11.0,4.0){\tiny4} \end{picture}\,\,\,, \Gamma=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 1} \put(6.0,3.0){\tiny\tiny 2} \end{picture}\in{\mathcal{G}}. Take F:=\Gamma\Gamma=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 1} \put(6.0,3.0){\tiny\tiny 2} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 1} \put(6.0,3.0){\tiny\tiny 2} \end{picture}\in M({\mathcal{G}}). Then $V(F)=\{1,2,1,2\}$. Now we define

[TABLE]

where $U\uplus V$ means that $U$ and $V$ can be expressed as

[TABLE]

with

[TABLE]

Further, the coproduct $\Delta_{G}$ can be extended to $H({\mathcal{G}})={\bf k}M({\mathcal{G}})$ by linearity. Let us compute some examples for better understanding of Eq. (5).

Example 2.8**.**

Let G=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny1} \put(11.0,-2.0){\tiny2} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny3} \put(6.0,3.0){\tiny4} \end{picture} and \Gamma_{2}\Gamma_{1}=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny3} \put(6.0,3.0){\tiny4} \end{picture}\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny1} \put(11.0,-2.0){\tiny2} \end{picture}\in M({\mathcal{G}}) with \Gamma_{1}=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(8.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(1,0){5.0}} \put(-2.0,-2.0){\tiny1} \put(11.0,-2.0){\tiny2} \end{picture}  ,  \Gamma_{2}=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny3} \put(6.0,3.0){\tiny4} \end{picture}. Then

[TABLE]

[TABLE]

and so

[TABLE]

[TABLE]

Further

[TABLE]

Example 2.9**.**

Consider G=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny3} \put(6.0,3.0){\tiny4} \end{picture}\begin{picture}(12.0,12.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(0.0,0.0){\line(0,1){5.0}} \put(0.0,5.0){\circle*{2.0}} \put(0.0,5.0){\line(0,1){5.0}} \put(0.0,10.0){\circle*{2.0}} \put(3.0,-2.0){\tiny1} \put(3.0,3.0){\tiny2} \put(3.0,9.0){\tiny5} \end{picture} and F=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny1} \put(6.0,3.0){\tiny2} \end{picture}\begin{picture}(12.0,12.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(0.0,0.0){\line(0,1){5.0}} \put(0.0,5.0){\circle*{2.0}} \put(0.0,5.0){\line(0,1){5.0}} \put(0.0,10.0){\circle*{2.0}} \put(3.0,-2.0){\tiny1} \put(3.0,3.0){\tiny2} \put(3.0,9.0){\tiny5} \end{picture}\in M({\mathcal{G}}). Then

[TABLE]

and

[TABLE]

Now we prove that the combinatorial definition of $\Delta_{G}$ in Eq. (5) is the same as the one given in Subsection 2.2. For this, it suffices to show that $\Delta_{G}$ in Eq. (5) satisfies Eqs. (2)—(4). For the cases of Eqs. (2) and (3), it follows directly from the definition of $\Delta_{G}$ in Eq. (5). For the case of Eq. (4), we have

Lemma 2.10**.**

Let $\Delta_{G}$ be given in Eq. (5) and $\Gamma_{1},\cdots,\Gamma_{n}\in{\mathcal{G}}$ with $n\geq 2$. Then

[TABLE]

Proof.

We have

[TABLE]

This completes the proof. ∎

So we conclude

Proposition 2.11**.**

The $\Delta_{G}$ given in Eq. (5) coincides with the $\Delta_{G}$ given in Subsection 2.2.

Remark 2.12**.**

The coproduct $\Delta_{G}$ in Eq. (5) is cocommutative.

2.4 A coalgebraic structure on subgraphs of a given graph

In this subsection, we obtain a coalgebra structure on subgraphs of a given graph $G$. Firstly, we define a linear map $\varepsilon_{G}:H({\mathcal{G}})\rightarrow{\bf k}$ by taking

[TABLE]

Lemma 2.13**.**

Let $G$ be a graph. Then the triple $(H({\mathcal{G}}),\,\Delta_{G},\,\varepsilon_{G})$ is a coalgebra.

Proof.

We first show the coassociativity

[TABLE]

By linearity, it suffices to consider basis elements $F\in M({\mathcal{G}})$. We have

[TABLE]

Thus $\Delta_{G}$ is coassociative. Next we prove the counicity of $\varepsilon_{G}$. For $F\in M({\mathcal{G}})$,

[TABLE]

With the same argument, we can show $({\rm id}\otimes\varepsilon_{G})\Delta_{G}=\beta_{r}$. Here $\beta_{l}$ and $\beta_{r}$ are the isomorphisms defined in Definition 2.5. This completes the proof. ∎

Corollary 2.14**.**

Let $G$ be a graph and $G^{\prime}$ be a subgraph of graph $G$. Then the triple $(H({\mathcal{G}}^{\prime}),\,\Delta_{G},\,\varepsilon_{G})$ is a subcoalgebra of the coalgebra $(H({\mathcal{G}}),\,\Delta_{G},\,\varepsilon_{G})$.

Proof.

It follows from the Lemma 2.13 and Eqs. (5) and (6). ∎

Corollary 2.15**.**

Let $G$ be a graph. Then $(H(\bullet),\,\Delta_{G},\,\varepsilon_{G})$ is a simple subcoalgebra, for any $\bullet\in V(G)$.

Proof.

It follows directly from the Corollary 2.14 and Eq. (5). ∎

2.5 A bialgebraic structure on subgraphs of a given graph

Combing the algebraic structure obtained in Subsection 2.1 and the coalgebraic structure obtained in Subsection 2.4, we build a bialgebraic structure on top of subgraphs of a given graph. Let us review the concept of bialgebras.

Definition 2.16**.**

1. (a)

[15, p. 49] Let $(H,\,m_{H},\,u_{H})$ and $(L,\,m_{L},\,u_{L})$ be two algebras. A map $f:H\rightarrow L$ is called an algebra morphism if

[TABLE] 2. (b)

[15, p. 51] A bialgebra is a quintuple $(H,\,m,\,u,\,\Delta,\,\varepsilon)$, where $(H,\,m,\,u)$ is an algebra and $(H,\,\Delta,\,\varepsilon)$ is a coalgebra such that $\Delta:H\rightarrow H\otimes H$ and $\varepsilon:H\rightarrow{\bf k}$ are morphisms of algebras.

Now, we arrive at our main result in this subsection.

Theorem 2.17**.**

Let $G$ be a graph. Then the quintuple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$ is a bialgebra.

Proof.

In Subsection 2.1, we know that the triple $(H({\mathcal{G}}),\,m_{G},\mathbbm{1})$ is an algebra. The triple $(H({\mathcal{G}}),\,\Delta_{G},\,\varepsilon_{G})$ is a coalgebra by Lemma 2.13. Further, the coproduct $\Delta_{G}$ is an algebra morphism by Eq. (4). We are left to prove that $\varepsilon_{G}$ is an algebra morphism. For this, let $F_{1},F_{2}\in M({\mathcal{G}})$. Then by Eq. (6)

[TABLE]

and

[TABLE]

Thus, $\varepsilon_{G}(F_{1}F_{2})=\varepsilon_{G}(F_{1})\varepsilon_{G}(F_{2})$ and $\varepsilon_{G}$ is an algebra morphism. This completes the proof. ∎

3 A Hopf algebra on subgraphs of a given graph

In this section, we construct a Hopf algebraic structure on subgraphs of a given graph $G$.

3.1 A Hopf algebraic structure on subgraphs of a graph

Let us review some concepts needed later.

Definition 3.1**.**

[1, p. 61] Let $(C,\,\Delta,\,\varepsilon)$ be a coalgebra and $(A,\,m,\,u)$ be an algebra. For $f,g\in{\rm Hom}(C,A)$,

[TABLE]

is said to be the convolution of $f$ and $g$.

Definition 3.2**.**

[1, p. 61] Let $(H,\,m,\,u,\,\Delta,\,\varepsilon)$ be a bialgebra. Then the triple $({\rm Hom}(H,H),\,\ast,\,u\circ\varepsilon)$ is a ${\bf k}$-algebra. A linear endomorphism $S$ of $H$ is called an antipode for $H$ if it is the inverse of ${\rm id}_{H}$ under the convolution product:

[TABLE]

A bialgebra with an antipode is called a Hopf algebra.

Definition 3.3**.**

[15, Definition 2.3.1] A bialgebra $(H,\,m,\,u,\,\Delta,\,\varepsilon)$ is called a graded bialgebra if there are ${\bf k}$-submodules $H^{(n)},n\geq 0$, of $H$ such that

1. (a)

$H=\bigoplus\limits^{\infty}_{n=0}H^{(n)}$; 2. (b)

$H^{(p)}H^{(q)}\subseteq H^{(p+q)},\,\,p,q\geq 0;$ 3. (c)

$\Delta(H^{(n)})\subseteq\bigoplus\limits_{p+q=n}H^{(p)}\otimes H^{(q)},\,\,n\geq 0,\,\,0\leq p,q\leq n.$

Elements of $H^{(n)}$ are said to have degree $n$. The $H$ is called connected if $H^{(0)}={\bf k}$ and $\ker\varepsilon=\bigoplus\limits_{n\geq 1}H^{(n)}$.

Lemma 3.4**.**

[2, 11, 15] Any connected graded bialgebra $H$ is a Hopf algebra. The antipode $S$ is given by:

[TABLE]

It is also defined by $S(1_{H})=1_{H}$ and recursively by any of the two formulas:

[TABLE]

Here $x^{\prime}$ and $x^{\prime\prime}$ are from

[TABLE]

We proceed to prove that $H({\mathcal{G}})$ is a connected graded bialgebra. For this, denoted by

[TABLE]

where $V(F)$ is the multiset of vertices of $F$. Now we arrive at one of our main results in this section.

Theorem 3.5**.**

Let $G$ be a graph. Then

1. (a)

The quintuple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$ is a connected graded bialgebra. 2. (b)

The quintuple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$ is a Hopf algebra with antipode $S=\sum\limits_{k\geq 0}(e-{\rm id}_{H})^{\ast k}$.

Proof.

(a). The quintuple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$ is a bialgebra by Theorem 2.17. By the definition of $H({\mathcal{G}})$, we obtain

[TABLE]

Let $F_{1}\in H({\mathcal{G}})^{(p)}$ and $F_{2}\in H({\mathcal{G}})^{(q)}$ with $p,q\geq 0$. Then

[TABLE]

whence

[TABLE]

By Eq. (5),

[TABLE]

Thus, the quintuple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$ is a graded bialgebra. Further,

[TABLE]

Therefore, the quintuple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$ is a connected graded bialgebra.

(b). It follows from Lemma 3.4 and Item (a). ∎

3.2 The dual Hopf algebra

In this subsection, we consider the dual Hopf algebra of $(H({\mathcal{G}}),\,m_{G},\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$. The following elementary result is fundamental.

Lemma 3.6**.**

[11] Let $H=\bigoplus\limits^{\infty}_{n=0}H^{(n)}$ be a graded bialgebra and $H^{(n)}$ be finite dimensional. Then

1. (a)

The graded dual $H^{\ast}$ is $\bigoplus\limits^{\infty}_{n=0}\left(H^{(n)}\right)^{\ast}$. Note that $H^{\ast}$ is also a graded bialgebra, and $H^{\ast\ast}\simeq H$. 2. (b)

$H\otimes H$ is also a graded bialgebra with $(H\otimes H)^{(n)}=\sum\limits_{i=0}^{n}H^{(i)}\otimes H^{(n-i)}$ for all $n\in\mathbb{N}$. Moreover, $(H\otimes H)^{\ast}\simeq H^{\ast}\otimes H^{\ast}$.

The quintuple $(H({\mathcal{G}}),\,m_{G},\,\mathbbm{1},\,\Delta_{G},\,\varepsilon_{G})$ is a connected graded bialgebra by Theorem 3.5 and $H({\mathcal{G}})^{(n)}$ is finite dimensional. Then its connected graded dual inherits also a graded Hopf algebra by Lemma 3.6, denoted by $(H({\mathcal{G}})^{\ast},\,\Delta_{G}^{\ast},\,\varepsilon_{G}^{\ast},\,m_{G}^{\ast},\,\mathbbm{1}^{\ast})$.

For each $F\in M({\mathcal{G}})$, we define

[TABLE]

where $\delta_{F,F^{\prime}}$ is Kronecker function. As \left\{Z_{F}{\,~{}{\tiny\left|\begin{array}[]{c}{}\hfil\\ {}\hfil\end{array}\right.\!\!\!\!\!\!\!}~{}\,}F\in M({\mathcal{G}})^{(n)}\right\} is a basis of $(H({\mathcal{G}})^{(n)})^{\ast}$, \left\{Z_{F}{\,~{}{\tiny\left|\begin{array}[]{c}{}\hfil\\ {}\hfil\end{array}\right.\!\!\!\!\!\!\!}~{}\,}F\in M({\mathcal{G}})\right\} is a basis of $H({\mathcal{G}})^{\ast}$.

The following result is a combinatorial description of the multiplication in $H({\mathcal{G}})^{\ast}$.

Theorem 3.7**.**

Let $G$ be graph and $F_{1},\,F_{2}\in M({\mathcal{G}})$. The product of $Z_{F_{1}}$ and $Z_{F_{2}}$ is given by

[TABLE]

where $n(F_{1},F_{2};F)$ is the number of $F_{1}\otimes F_{2}$ in $\Delta_{G}(F)$.

Proof.

Let $G$ be a graph. We consider the basis elements of $H({\mathcal{G}})^{\ast}$. For any $F_{1},\,F_{2}\in M({\mathcal{G}})$, we suppose

[TABLE]

For any $F\in M({\mathcal{G}})$, we have

[TABLE]

and

[TABLE]

Thus $c_{F_{1},F_{2}}^{F}=n(F_{1},F_{2};F)$. This completes the proof. ∎

Example 3.8**.**

Let $G$ be a graph in Item (c) of Example 2.4. We consider the product of $H({\mathcal{G}})^{\ast}$.

1. (a)

Let F_{1}=\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture} and F_{2}=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 2} \put(6.0,3.0){\tiny\tiny 1} \end{picture}. Then

[TABLE] 2. (b)

Let F_{1}=\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 3} \end{picture} and F_{2}=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 2} \put(6.0,3.0){\tiny\tiny 1} \end{picture}. We have

[TABLE]

The coproduct $m_{G}^{\ast}$ of $H({\mathcal{G}})^{\ast}$ is given by

Lemma 3.9**.**

Let $G$ be graph and $\Gamma_{1}\cdots\Gamma_{n}\in M({\mathcal{G}})$. The coproduct of $Z_{\Gamma_{1}\cdots\Gamma_{n}}\in M({\mathcal{G}})^{\ast}$ is given by

[TABLE]

with the convention that $Z_{\Gamma_{1}\Gamma_{0}}=\mathbbm{1}$ and $Z_{\Gamma_{n+1}\Gamma_{n}}=\mathbbm{1}$.

Proof.

We consider the basis elements of $H({\mathcal{G}})^{\ast}$. Suppose

[TABLE]

For any $F_{1},F_{2}\in M({\mathcal{G}})$, we have

[TABLE]

and

[TABLE]

Thus, $c_{F_{1},F_{2}}=1$ if $\Gamma_{1}\cdots\Gamma_{n}=F_{1}F_{2}$ and $c_{F_{1},F_{2}}=0$ otherwise. This completes the proof. ∎

Example 3.10**.**

Let $G$ be a graph in Item (c) of Example 2.4. We consider the coproduct of $H({\mathcal{G}})^{\ast}$. Let \Gamma_{1}=\begin{picture}(10.0,5.0)(-2.0,-1.0)\put(0.0,0.0){\circle*{2.0}} \put(3.0,-2.0){\tiny 1} \end{picture} and \Gamma_{2}=\begin{picture}(12.0,5.0)(0.0,-1.0)\put(3.0,0.0){\circle*{2.0}} \put(3.0,0.0){\line(0,1){5.0}} \put(3.0,5.0){\circle*{2.0}} \put(6.0,-2.0){\tiny\tiny 2} \put(6.0,3.0){\tiny\tiny 1} \end{picture}. Then

[TABLE]

3.3 The algebra morphisms induced from graph homomorphisms

In this subsection, we are going to consider algebra morphisms induced by graph homomorphisms. For this, let us recall the concept of a graph homomorphism.

Definition 3.11**.**

[19, p. 4] Let $G_{1}$ and $G_{2}$ be two graphs. A homomorphism of $G_{1}$ to $G_{2}$, written as $f:G_{1}\rightarrow G_{2}$, is a mapping $f:V(G_{1})\rightarrow V(G_{2})$ such that $(f(u),f(v))\in E(G_{2})$ whenever $(u,v)\in E(G_{1})$.

Next, we give the main result in this subsection. Let $f:G_{1}\rightarrow G_{2}$ be a graph homomorphism. Then $f$ can be restricted to any subgraphs of $G_{1}$, still denoted by $f$. We define the linear map $H(f):H({\mathcal{G}}_{1})\rightarrow H({\mathcal{G}}_{2})$ given by

[TABLE]

and

[TABLE]

Theorem 3.12**.**

The map $H(f):H({\mathcal{G}}_{1})\rightarrow H({\mathcal{G}}_{2})$ is an algebra morphism.

Proof.

Let

[TABLE]

Then by Eq. (10),

[TABLE]

Thus $H(f)$ is an algebra morphism. This completes the proof. ∎

As an application of Theorem 3.12, we obtain a functor from the graph category to the algebra category. Let us recall

Definition 3.13**.**

[21, Definition 7.1] A **category ** is a class $\mathfrak{C}$ of objects (denoted $A,B,C,\cdots$) together with

1. (a)

a class of disjoint sets, denoted $\hom(A,B)$, one for each pair of objects in $\mathfrak{C}$; (an element $f$ of $\hom(A,B)$ is called a morphism from $A$ to $B$ and is denoted $f:A\rightarrow B$); 2. (b)

for each triple $(A,B,C)$ of objects of $\mathfrak{C}$ a function:

[TABLE]

(for morphisms $f:A\rightarrow B$, $g:B\rightarrow C$, this function is written $(g,f)\mapsto g\circ f$ and $g\circ f:A\rightarrow C$ is called the composite of $f$ and $g$); all subject to the two axioms:

1. (i)

Associativity. If $f:A\rightarrow B$, $g:B\rightarrow C$, $h:C\rightarrow D$ are morphisms of $\mathfrak{C}$, then $h\circ(g\circ f)=(h\circ g)\circ f$. 2. (ii)

Identity. For each object $B$ of $\mathfrak{C}$ there exists a morphism $1_{B}:B\rightarrow B$ such that for any $f:A\rightarrow B$, $g:B\rightarrow C$,

[TABLE]

Definition 3.14**.**

[21, Definition 1.1] Let $\mathfrak{C}$ and $\mathfrak{D}$ be categories. A covariant functor T from $\mathfrak{C}$ to $\mathfrak{D}$ (denoted $T:\mathfrak{C}\rightarrow\mathfrak{D}$) is a pair of functions (both denoted by $T$), an object function that assigns to each object $C$ of $\mathfrak{C}$ an object $T(C)$ of $\mathfrak{D}$ and a morphism function which assigns to each morphism $f:C\rightarrow D$ of $\mathfrak{C}$ a morphism

[TABLE]

of $\mathfrak{D}$, such that

1. (a)

$T(1_{C})=1_{T(C)}$ for every identity morphism $1_{C}$ of $\mathfrak{C}$; 2. (b)

$T(g\circ f)=T(g)\circ T(f)$ for any two morphisms $f,g$ of $\mathfrak{C}$ whose composite $g\circ f$ is defined.

Let $\mathfrak{G}$ be the class of all graphs, for $G_{1},G_{2}\in\mathfrak{G}$, $\hom(G_{1},G_{2})$ is the set of all graph homomorphisms $f:G_{1}\rightarrow G_{2}$. Then $\mathfrak{G}$ is a category, called the graph category. Let $\mathfrak{H}$ be the ${\bf k}$-algebra category whose objects are all ${\bf k}$-algebra; the $\hom(A_{1},A_{2})$ is the set of all ${\bf k}$-algebra morphisms $\varphi:A_{1}\rightarrow A_{2}$.

Now we arrive at out main result of this section.

Theorem 3.15**.**

Let $\mathfrak{G}$ be the graph category and $\mathfrak{H}$ the ${\bf k}$-algebra category. Define

[TABLE]

where $G$ is a graph in $\mathfrak{G}$, $f:G_{1}\rightarrow G_{2}$ is a graph homomorphism and $H(f):H({\mathcal{G}}_{1})\rightarrow H({\mathcal{G}}_{2})$ is the algebra morphism obtained in Theorem 3.12. Then $H(-)$ is a covariant functor.

Proof.

By Theorem 3.12, it suffices to prove Items (a) and (b) in Definition 3.14. For Item (a), let $F=\Gamma_{1}\cdots\Gamma_{n}\in M({\mathcal{G}})$. Then by Eq. (10),

[TABLE]

For Item (b), let

[TABLE]

be graph homomorphisms. For $F=\Gamma_{1}\cdots\Gamma_{n}\in M({\mathcal{G}}_{1})$,

[TABLE]

where the third equation employs the fact that the image of a connected subgraph of a graph homomorphism is a connected subgraph. Thus $H(g)\circ H(f)(F)=H(g\circ f)(F)$. This completes the proof. ∎

Acknowledgments: This work was supported by the National Natural Science Foundation of China (Grant No. 11571155, 11771191, 11861051). We thank the anonymous referee for valuable suggestions helping to improve the paper.