A Note Concerning Equipotent Digraph Homomorphism Sets

Functor adjunctions are fundamental to category theory and have recently found applications in the empirical sciences. In this paper a functor adjunction on a special full subcategory of the category of digraphs is borrowed from mathematical biology and used to equate cardinalities of sets of homomorphisms between various types of digraphs and associated line digraphs. These equalities are especially useful for regular digraphs and are applied to obtain homomorphism set cardinality equalities for the classes of de Bruijn digraphs and Kautz digraphs. Such digraphs play important roles in bioinformatics and serve as architectures for distributed high performance computing networks.


Introduction
Every area of mathematics (e.g., group theory, topology) is described by numerous definitions, theorems, and constructions. However, many common mathematical concepts occur naturally with only slight variation in these various areas of mathematics. Category theory is that branch of mathematics which identifies and studies such common concepts and provides formal mechanisms for mapping them from one area of mathematics to another. More specifically, a category (e.g., the category of sets) consists of a class of objects (e.g., sets), morphisms between objects (e.g., maps between sets), an identity morphism for each object (e.g., the set identity map), and a rule for associatively composing morphisms (e.g., composition of maps). Functors provide formal maps between categories (e.g., from the category of groups to the category of sets) by associating objects and morphisms in different categories subject to the constraints that morphism composition and object identities are preserved.
Although this adjoint pair of functors has important consequences for biological systems, the objective of this paper is to exploit HG's functor adjunction for the purpose of establishing equalities between the cardinalities of homomorphism sets for various types of digraphs and associated line digraphs. These equalities are especially useful for regular digraphs and they are applied to obtain results for the classes of de Bruijn digraphs and Kautz digraphs. Both de Bruijn and Kautz digraphs play important roles in bioinformatics (e.g., [17]) and serve as efficient architectures for distributed high performance computing networks (e.g., [18]). To make this paper relatively self-contained, basic graph and category theoretic definitions that are used in this paper are summarized in the next section and results from HG's research that are relevant to this work are presented in section 3. The theory developed in section 3 is used in section 4 to establish this paper's main theorems concerning equalities between http://www.ijSciences.com May 2019 (05) -Volume 8 48 equipotent digraph homomorphism sets. These main results are applied to dicycles, de Bruijn digraphs, and Kautz digraphs in section 5. Closing remarks comprise the final section of this paper.

Basic Definitions
where is a set of arcs, is a set of nodes, is a map that sends each arc to its source node, and is a map that sends each arc to its target node. A source (target) node is adjacent to (from) a target (source) node and the in-degree (out-degree) ( ) ( ( )) of a node is the number nodes adjacent to (from) . is -regular if ( ) ( ) for each of its nodes. If is a partition of the node set of into non-empty subsets , then ⁄ is the associated quotient digraph of with node set and arc set {( ) ( ) }.
The line digraph of digraph ( ) is the digraph which has as its node set and arc set ( ) . A morphism is an isomorphism if there exists a morphism such that and . A category is a full subcategory of category if every object of is an object of , ( ) ( ) for all objects and in , for every object in the identity morphism is the same in as it is in , and the composite of two morphisms in is the same as their composite in .
Examples of categories are the category (where is the collection of all sets, the morphisms are the ordinary mappings between sets, and is the usual composition of maps), the category (where is the collection of all groups, the morphisms are the ordinary group homomorphisms, and is the usual composition of group homomorphisms), and the category of digraphs (where is the collection of all digraphs, the morphisms are digraph homomorphisms, and is composition of digraph homomorphisms). It is easily verified that , , and Dgp satisfy items (1), (2a), and (2b) above.
Functors can be regarded as morphisms between categories and -in a sense -they provide a "picture" of what one category looks like inside another. If is a covariant functoror simply a functor (contravariant functors are not used here) -from category to category (denoted ), then it assigns to every in an object in and to every ( ) an ( ) such that: (3) for every in ; and (4) when is defined in , then is defined in and ( ) .
If is a functor such that , then is an endofunctor.
Simple examples of functors are the identity functor (which makes the assignments for every in and for every ( )) and the forgetful functor (which assigns to every group in its underlying set in and to each homomorphism ( ) the set map ( ) -i.e., forgets the group structure going from to ). It can be determined by inspection that these functors satisfy the required properties given above by items (3) and (4). It is also straightforward to see that if and are functors, then their composition is also a functor.

The Endofunctors and on Category and Their Adjunction
The results found in HG's research that are required to provide the main results of this paper are presented in this section. However, it should be noted that HG omitted in [15,16] explicit proofs that their "transformation operators" and are endofunctors. For the sake of completeness, the proofs that and are endofunctorsalong with appropriate homomorphism definitions -are developed in this section. Also note that for the purpose of mnemonics, HG's ( ) is the ( ) used herein.
Fundamental to this note is the category defined by HG which has as its objects digraphs ( ) with the property that for every there exists such that and has as its morphisms the associated digraph homomorphisms. The following lemma follows trivially from the definition of full subcategory and is stated without proof.

Lemma 1.
is a full subcategory of Dgp.

Lemma 6. is an endofunctor on .
Proof of Lemma 6. That is an object in when is an object in is shown to be true by Proposition 5.7 (iii) in [15]. To show that item (3)  ) .

Lemma 7.
. Proof of Lemma 7. This is shown to be true in Theorem 5.9 in [15].
is called here the digraph return functor on sincefrom Lemma 7 -, i.e. the action of upon the line graph of "returns" to an isomorphic copy of itself in . Denote this isomorphism by .

Lemma 8.
, that is ( ) ( ) for all objects and in . Proof of Lemma 8. This is a restatement of Theorem 5.8 in [15]. □

Main Results
The above lemmas lead to the following main results which equate the cardinalities of homomorphism sets between digraphs, line digraphs, and associated quotient digraphs.  But from Lemma 7, so that this now becomes ∑ ( ⁄ ) ∑ ( ⁄ ) .
Since , then ⁄ ⁄ . The result follows from the fact that there is a one-to-one correspondence between the set of injective homomorphisms from ⁄ into and the set of injective homomorphisms from ⁄ into .

Theorem 12. If is an object in , then
( ) ( ). Proof of Theorem 12. Let in Theorem 9 in which case | ( )| | ( )| ( ). But from Lemma 7, so that now there is a one-to-one correspondence between the morphisms in set ( ) and those in ( ). Consequently, | ( )| ( ) and the proof is complete.

Corollary 13. is rigid if, and only if,
is rigid. Proof of Corollary 13. If is rigid, then ( ) ( ) which implies that is also rigid. If is rigid, then ( ) ( ) which implies that is also rigid. □

Dicycles
It is clear that the dicycles (directed cycles) ⃗ on nodes are objects in since they are 1-regular, i.e. for every node in ⃗ there is exactly one arc and exactly one arc such that . Dicycles are useful here because they provide trivial validations for aspects of the main results. In particular, since it is well known that ⃗ ⃗ , then substituting this identity into and setting ⃗ in

de Bruijn Digraphs
Let and be the set of all strings on of length and recall that the de Bruijn digraph ( ) of dimension on symbols has as its node set with an arc in ( ) if and only if when . Because -as is well known -( ) is -regular, de Bruijn digraphs are objects in . It is also well known that ( ) is the line digraph of ( ), provided that both digraphs have the same symbol set . These properties lead to the following results for de Bruijn digraphs: the de Bruijn subdigraphs and since they areregular, they too are objects in . It is also well known that for a fixed symbol set , ( ) is the line digraph of ( ). Since the proofs of the following theorems for Kautz digraphs closely follow those for de Bruijn digraphs, they are stated without proof. ( ( )) ( ( ))

Closing Remarks
The results in this paper were obtained via a novel application of a functor adjunction borrowed from theoretical biology. This approach not only illustrates the utility of category theory, but also suggests that category theory might play a role in acquiring in a fairly straightforward manner interesting new general mathematical results from otherwise unexpected disparate areas of science.
In closing, it is noted that -whereas digraphs model system topologiestheir associated line digraphs model what might be called their associated "interstitial spaces". Consequently, the main results of this paper might be useful during aspects of engineering design processes. For example, if a system's topology is represented by a digraph in category , it might be important to know from Theorem 12 that a system and its "interstitial space" can be homomorphically "collapsed" onto themselves the same the number of ways.