A survey of digraph invariants

A Survey of directed graphs invariants Sheng Chen, Yilong Zhang schen@hit.edu.cn, zhangyl_math@hit.edu.cn Dept. of Mathematics, Harbin Institute of Technology, Harbin 150001, P.R.China

Also available on arXiv(1412,4599; v2) Abstract. In this paper, various kinds of invariants of directed graphs are summarized. In the first topic, the invariant đ?‘¤(đ??ş) for a directed graph G is introduced, which is primarily defined by S. Chen and X.M. Chen to solve a problem of weak connectedness of tensor product of two directed graphs. Further, we present our recent studies on the invariant đ?‘¤(đ??ş) in categorical view.

In the second topic, Homology theory on directed graph is introduced, and we also cast on categorical view of the definition.

The third topic mainly focuses on Laplacians on graphs, including traditional work and latest result of 1-laplacian by K.C.Chang.

Finally, Zeta functions and Graded graphs are introduced, inclduing Bratteli-Vershik diagram, dual graded graphs and differential posets, with some applications in dynamic system.

Content 0. Introduction 1. The invariant đ?‘¤(đ??ş ) and weak connectedness of digraphs tensor product 1.1 A survey of historic work 1.2 A categorical view of đ?‘¤ (đ??ş ) 2. Homology theory on directed graphs 2.1 Homology on directed graphs 2.2 Remark: A categorical view 3. Laplacians on graphs 2.1 A survey of classical results 2.2 Recent studies of 1-laplacian on graphs by K. C. Chang 4.

Zeta functions

5.

Graded graphs 5.1 Graded graphs and differential posets 5.2 Quatized dual graded graphs, Bratteli-Vershik diagram

6.

References

1

0. Introduction.

We shall introduce the notation and terminology below on directed graphs. These definitions are primarily prepared for the first topic.

A directed graph (abbreviated as digraph) đ??ş is a pair đ??ş ( ) , with . For an arc ( ) , u is called initial vertice of the arc and v is called end vertice; multiple arcs are arcs with same starting and ending nodes. { } and The directed cycle is the digraph with vertice set arc set {( ) ( ) ( )}. In particular, consists of a single vertice with a loop. The directed path is with the arc ( ) removed. The initial vertice of is and the end vertice of is . The length of a directed cycle or path is defined to be the number of arcs it contains.

A route of length n in a digraph đ??ş is a pair (S, ), where ( ) is a series of vertices, and ( ) is the orientation of corresponding edges with ( ) , and ( ) , . A route is closed if ; a closed route is unrepeated means whenever , except for a route is open if The weight of route R is ( ) ∑ ( ) defined Let ( ) be a route with ( ) and ( ); let ( ) be another route with ( ) and ( ) . If , the composite route ( ) ( ) is defined to be ( ).

A walk is a route with ( ), there are three kinds of connectedness for digraphs defined as follow. A digraph đ??ş is strong (strongly connected) if for each pair , there is a walk from u to v; it is unilateral (unilaterally connected) if for each pair , there is a walk from u to v or a walk from v to u; it is weak (weakly connected) if for each , there is a route ( ) starts at u and ends at v. The diameter of digraph (đ??ş) is defined to be the maximal weight of routes in đ??ş i.e. (đ??ş ) {đ?‘¤( )| }.

Let đ??ş đ??ş be digraphs. Their tensor product đ??ş đ??ş is defined to be the ( ) digraph with vertice set , and ( ( )) is an arc in đ??ş đ??ş if ( ) and ( ) are arcs in đ??ş and đ??ş .

A digraph map đ??ş đ??ş is a homomorphism if , and , where are maps attach each edge to its initial vertice and end vertice, respectively. is said to be epimorphism if it is both a homomorphism and a surjevtive.

1, The invariant

( ) and weak connectedness of digraphs

tensor product 2

1.1 A survey of historic work Harary [1] raised problems to characterize the digraphs whose tensor product has each of the three kinds of connectedness, i.e. strong, unilateral, and weak. Harary [1] solved unilateral problem in 1966; McAndrew [2] solved strong one in 1963; and Chen [3] solved weak one in 2014.

To solve problem of tensor product with strong connectedness, McAndrew introduced invariant (đ??ş ) for digraph G, where (đ??ş ) ( ( )) is first introduced, where C takes over all of directed cycles of digraph đ??ş ( ), and n(C) denote the length of cycle C. This definition captures the idea of the “distanceâ€? between two points in G, in the sense of modulo (đ??ş). Further, tensor product with strong connectedness can be characterized by the invariant (đ??ş ) Theorem 1.1 (McAndrew) đ??ş are strong, and

( (đ??ş )

đ??ş is strong connected if and only if đ??ş and đ??ş (đ??ş ))

.

To solve problem of weak connectedness, S. Chen and X.M. Chen introduced invariant đ?‘¤(đ??ş ), where đ?‘¤ (đ??ş ) ( ( )), where R is taking all of the routes of đ??ş, and the definition of route can be found in Introduction. đ?‘¤ (đ??ş ) is called the weight of G. To compare the two definitions, one may notice that đ?‘¤(đ??ş ) is a generalization of (đ??ş ), because a directed cycle is always a route. Therefore (đ??ş )|đ?‘¤(đ??ş ), if we assume | in some cases.

Theorem 1.2 (S. Chen, X. M. Chen) Let both đ??ş and đ??ş are weak connected, đ?‘¤ (đ??ş )

đ??ş is weak connected if and only if (đ?‘¤(đ??ş) đ?‘¤(đ??ş )) and one of đ??ş containing sources (sinks) implies other contains no sinks (sources).

(1) If

(i=1,2), đ??ş

(2) If one of đ??ş , say đ??ş , has weight zero, then đ??ş (đ??ş ) đ??ş is l-chainable, where N,

đ??ş is weak if and only if

Remark: đ??ş is l-chainable if there does not exist a pair of permutation matrix M, (đ??ş) being the adjacent matrix of đ??ş, such that

ďƒŠ A 0ďƒš MA(G ) N  ďƒŞ 1 ďƒş.

0 A ďƒŤ 2ďƒť

1.2 A categorical view of

( )

Our aim is to characterize the weight đ?‘¤(đ??ş ) through digraph homomorphism.

Our result is: for digraph G with positive weight, there is an epimorphism from G to ( ) ; for digraph G with positive weight, there is an epimorphism from G to ( ) Further, there is a functor from category of digraphs with positive w(G) to category 3

of directed circles, there is a functor from category of digraphs with zero w(G) to category of directed path.

Before coming to weight of digraph, we first study some properties of weight of routes ( ). In this section, we always assume that a digraph is weakly connected and has no multiple edges, and the vertice set is finite.

Lemma 1.4 Let to be the composition of routes, route, and be the digraph homomorphism, one has

be the inverse of

(1) ď ł ( R1R2 )  ď ł ( R1 )  ď ł ( R2 ); (2) ď ł ( R1 )  ď€­ď ł ( R); (3) ď ł (ď Ş ( R))  ď ł ( R).

The formula (1) implies is additive; and formula (2) implies if the weight of is negative, its inverse is positive. As and serve the same role in calculation, we always assume that the weight of is positive. Formula (3) is true since ( ) ( ( ) ( )), where ( ) ( ( ) ( ) ( )), and ( ) .

Next, we reformulate the definition of weight as follow.

Definition 1.3 The weight đ?‘¤(đ??ş ) ( ) is defined for each digraph G, { | ( ) }, i.e. n belongs to where if there is a closed route R in G such that ( ) . If is an empty set, we define đ?‘¤(đ??ş ) An immediate question rises up: How to calculate đ?‘¤ (đ??ş ) for a certain digraph?

In Definition2, the greatest common divisor is taken over weight of all closed route in G, there can be many of routes. Actually, if is non-empty, it is infinite. To see that, if there is a closed route R, s.t ( ) , then ( ) , for each , therefore W contains an infinite set and W itself is infinite.

However, one notes that

{ (

) (

)}

{ (

) (

)

(

)}

{ ( ) ( )} composed routes do NOT provide independent information of weights apart from routes and ; so they could be excluded from the calculation.

Now we are ready to present a theorem for calculating đ?‘¤(đ??ş ) Theorem 1.5 (Theorem of Calculating ( )) (Y.L.Zhang) đ?‘¤ (đ??ş ) ( ), { | ( ) } , Since the set of where vertices (đ??ş) is finite, is finite, therefore đ?‘¤(đ??ş ) can be calculated(through we have not yet found a combinatorial way).

The following to theorem is our first characterization of đ?‘¤(đ??ş ) Theorem 1.6 (Y.L.Zhang) For a digraph G with đ?‘¤ (đ??ş ) . There is an epimorphism đ??ş ( ) Conversely, let S be the set of positive integer n such 4

that there is an epimorphism from G to integer of S equal to đ?‘¤ (đ??ş ).

, then S is a finite set, and the largest

Proof: (1) We shall denote the vertices of }. Suppose ( ) as {0,1,2,‌, đ?‘¤ (đ??ş ) đ?‘¤ (đ??ş ) for G, then for an arbitrarily fixed vertice (đ??ş), set ( ) and ( ) for đ??ş define

ď Ş : V (G) ď‚Ž V (Cw(G ) ) q ď‚Ž ď ł ( R( p, q)) mod( w(G )) where ( ) is an arbitrary route start at p and end at q, this is well-defined since ( ) ( ) is a closed route in G, so for another route ( ). Since đ?‘¤(đ??ş )| ( ( (

(

) (

))

( (

)). We have ))

(

(

( ( ) (

))

( (

))

))

( (

(

(

))

))

(đ?‘¤(đ??ş))

The behavior of on edges (đ??ş) is uniquely determined by its behavior on (đ??ş), therefore is a homomorphism. Since is obviously a surjective, is indeed an epimorphism.

(2) We shall only prove that for đ?‘¤(đ??ş ) there cannot be a homomorphism from G to . If it is possible, and đ??ş is a digraph homomorphism, by definition of weight, there exists a route R in G, such that n is not a divisor of ( ) We shall show that the induced homomorphism | leads to contradiction.

To do this, we shall induce collapse of route. Graphically, 1

v j 1

1

-1

v j 1

vj

v j 2

1

1

v j 3 vi1

-1

1

vi1

vi

collapse

v j 1

1

vj

1

vi2

1

vi3

collapse

v j 3

vi1

1

vi

1

vi3

Fig 1.1 the collapse of route

Formally, in a closed route R, if there is a pair of adjacent edges has opposite orientations, i.e. , and , or and , a collapse of R is a closed route by deleting edges and from R, or deleting edges and according to the orientations; deleting ; then identify and .

Collapse induces digraph homomorphism . We repeat collapse recursively and the process shall terminate at step k with ( ) , for some integer k, since is of finite length and its weight is invariant under the collapse, i.e. ( ) ( ) ( ) Therefore,

( )(

)

is a digraph homomorphism. However, n is not a 5

⎕

( ), this is impossible.

divisor of

Theorem 1.7 (Y.L.Zhang) Suppose đ?‘¤ (đ??ş ) unique epimorphism

đ??ş

for a digraph G. There is a

( ).

Proof: let p be the initial point of the path with largest distance in G, set ( ) be the initial point of , where we denote the vertices of }. ( ) as {0,1,2,‌, Suppose đ?‘¤ (đ??ş ) for G, then for an arbitrarily fixed vertice (đ??ş) , set ( ) (đ??ş ) define and for

ď Ş : V (G) ď‚Ž V ( Pm ) q ď‚Ž ď ł ( R( p, q)) where ( ) is an arbitrary route start at p and end at q, this is well-defined, since ( ) ( ) is a closed route in G, so for another route ( ). Since (

(

) (

( (

))

)) ( (

In the same way,

. We have ))

(

(

( ( ) (

)) ))

( ( ( (

))

(

(

))

))

is a homomorphism, and a surjective. Therefore,

is ⎕

indeed an epimorphism. The uniqueness is obvious.

Lemma 1.8 (1) if

| , and

is a homomorphism, then

(2) if is a homomorphism, then monomorphism;

is an epimorhism; , and if

(3) if is a homomorphism, then , and if isomorphically and identify the initial and terminal point of (4) there is no homomorphism from

to

, ,

is a

maps edges

.

Corollary 1.9 (1) If

is a homomorphism between two closed routes, then

(2) If

is a homomorphism between two open routes, then

( )| ( ). ( )

( ).

Proof: This proof is similar to the Theorem1’s, we shall only prove (1) here. Suppose ( ) is not a divisor of ( ) Let to be the collapsed closed route of , and to be the induced closed route by collapsing corresponding edges and vertices as well as identifying vertices in the natural way, so that | is a homomorphism. Repeat collapse finite times, one will get (

)

( )

, and

, since weights are invariant under collapse operation. Therefore 6

|

( )

(

⎕

is a homomorphism, which is contradictory to Lemma2.

)

Corollary 1.10 (1) đ?‘¤(đ??ş )

(i=1,2). If

đ??ş

đ??ş is an epimorphism, then đ?‘¤(đ??ş )|đ?‘¤(đ??ş ).

(2) đ?‘¤(đ??ş )

(i=1,2). If

đ??ş

đ??ş is an homomorphism, then

(đ??ş )

(đ??ş ).

Proof: (1) If đ?‘¤(đ??ş ) is not a divisor of đ?‘¤(đ??ş ), there exists a closed route in đ??ş , such that for any route in đ??ş ( ) does not divide ( ), this is contradictory to Corollary1 (1), since

⎕

is a epimorphism. Proof of part (2) is similar.

Main Theorem 1.11 (Y.L.Zhang) (1) đ?‘¤ (đ??ş )

đ??ş

(i=1,2). If

epimorphism, Ěƒ

(

)

(

đ??ş is an epimorphism, and then ),

induce an

making the diagram1 commutes. In other words,

let Digraphw>0 to be the category of directed graphs with positive weights, C to be the category of directed circles, then : Digraphw>0 C is a functor. ď š

G1 ď Ş

ď Ľ

G1

G2

ď Ş

ď Ş

ď Ş

ď Ľ

ď š Cw(G1 )

G2

Pd (G1 )

Cw(G2 ) Diagram1,

Pd (G2 )

Diagram2

Fig1.2 Commutative diagrams for

(2) đ?‘¤ (đ??ş )

(i=1,2). If

epimorphism Ěƒ

(

)

đ??ş (

đ??ş is an homomorphism, and then ),

induce an

making the diagram2 commutes. In other words, let

Digraphw=0 to be the category of directed graphs with zero weight, P to be the category of directed circles, then : Digraphw=0 P is a functor.

Proof: (1) We shall define

ď š : Cw ( G ) ď‚Ž C w ( G ) 1

2

ď Ş ( p) ď‚Ž ď Ş ď š ( p) First, Ěƒ is well-defined: take then

, for

(

) in đ??ş ,

homomorphism, by Lemma1, one has ( (

( )

(

( (

))

( ( (

)) is a route in đ??ş which starts at 7

)

if

đ??ş

and

đ?‘¤(đ??ş ) Since )))

( (

( ), and ends at

))

( )

( ),

is a digraph đ?‘¤(đ??ş ) and ( ). Therefore,

đ?‘¤(đ??ş )| ( (

)). We have ď Ş ď š ( p)=ď Ş ď š (q) , so Ěƒ is well-defined.

Secondly, Ěƒ is a homomorphism. To prove this, one note that Ěƒ ( ( ) Ěƒ ( ( ))

)

, therefore Ěƒ maps edges to edges, and is indeed a homomorphism, and ⎕

then obviously a epimorphism.

2. Homology and Cohomology on digraphs Grigor’yan, Lin, Muranov, and S-T. Yau [4~6] systematically introduce the Homology and Cohomology theory on finite digraphs, based on the theories of p-path and p-form on finite sets, respectively. The boundary operator acting from p-path to (p-1)-path and derivative operator acting from p-form to (p+1)-form are naturally defined, satisfying In this section, we briefly introduce Homology theory on digraphs, and present our result based on the definition. As for the Cohomology theory on digraphs, it is dual to Homology case.

2.1 Homology on directed graphs Let K be a fixed field, and đ??ş

(

) be a digraph with the finite set V of

vertices. An elementary p-path on V is any (ordered) sequence i0 ,..., i p of p+1 of vertices that will be denoted by ei0 ,...,i p . Clearly,

ď ťe ď ˝ is a basis. Consider the free i0 ,...,i p

K-module ď Œ p  ď Œ p (V ) which is generated by elementary p-paths ei0 ,...,i p , whose elements are called p-paths, therefore each p-path has the form v

ďƒĽv

i0 ,...,i p

ei0 ,...,i p

i0 ,...,i p

Definition 2.1 The boundary operator

: ď Œ p ď‚Ž ď Œ p1 is defined by

p

ď‚śei0 ,...,i p  ďƒĽ (1)q e q 0

where ei0 ,...,i p is elementary p-path.

Note that the Definition 1. is equivalent to:

8

ďƒ™

i0 ...iq ...i p

(2.1)

(ď‚śv) 0

i ,...,i p1

p

 ďƒĽďƒĽ (1) q v 0

i ...iq1kiq ...i p1

.

kďƒŽV q 0

The product of any two paths u ďƒŽ ď Œ p and v ďƒŽ ď Œ q is defined as uv ďƒŽ ď Œ pq1 as follow:  u 0 pv 0

i ...i p j0 ... jq

i ...i

(uv) 0

j ... jq

.

(2.2)

For example, if u  ei0 ,...,i p and v  e j0 ,..., jq , then

ei0 ,...,ip e j0 ,..., jq  ei0 ,...,i p j0 ,..., jq .

Based on Definition 1, one can easily observe that Theorem 2.2  2  0 .

Theorem 2.3 For u ďƒŽ ď Œ p and v ďƒŽ ď Œ q , we have ď‚ś(uv)  (ď‚śu)v  (1) p1 (ď‚śv).

(2.3)

These basic results come parallelly from the Simplicial homology theories, and the only difference is that the bases are indexed by vertices set.

An elementary p-path ei0 ,...,i p is called regular if ik ď‚š ik 1 for all k, and let R p denote the submodule generated by regular p-path. Then a regular path is allowed if ik ik 1 ďƒŽ E, k  0,..., p  1 . Let Ap denote the submodule generated by all allowed

p-paths, one has Ap ď‚Ł Rp ď‚Ł ď Œ p .

However, the operator is not necessarily invariant on đ?‘? , for example in Fig2, đ?›Ź , but since 14 is not an edge of the digraph.

4 4 4 4 ∉ Therefore, to construct homology on digraph, we need to consider a submodule of đ?‘? which is -invariant.

Fig2.1 path

2

4

1

3

4

is not

9

-invariant

ď ť

ď ˝

Consider the submodule ď — p  ď — p  v ďƒŽ Ap , ď‚śv ďƒŽ Ap1 . By definition, ď — p is ď‚ś -invariant and its elements are called well-defined chain-complex: ď‚ś

-invariant p-paths, and by Theorem 1, one has

ď‚ś

ď‚ś

p1 p p1 ď‚žď‚ž ď‚ž ď‚Žď — p ď‚žď‚ž ď‚Žď — p1 ď‚žď‚ž ď‚ž ď‚Ž

ď‚ś0 ď‚ś2 ď‚ś1 ď‚žď‚ž ď‚Žď —1 ď‚žď‚ž ď‚Žď —0 ď‚žď‚ž ď‚Ž 0.

It follows that Definition 2.4 The homology of a digraph G=(V,E) is defined to be : H p (G)  Ker p / Im p1

(2.4)

Next, a question rises up: which sort of digraphs have non-trivial homology groups, and what is the necessary condition to have non-trivial homology groups ?

The simplest and most important examples of digraphs are triangle and square, with orientation defined in Fig3.

2

1

3

2

4

1

3

Fig2.2.(1) triangle

(2) square

The first interesting result is that they are the only closed route with trivial .Here closed route is a digraph with series of distinct vertices ( ), with or . We rephrase the result at hear: if

Theorem 2.5 Given a close route R, then either ( ) , R is triangle or square.

( )

or

To see why s of triangle and square are trivial, one notice that is closed, that is ( ) , and { } { } , and ( ) As for square, notice that ( 4 4) ( ) , so 4 . In the same way that

( )

. And

therefore , so 4

We end with another theorem which describes in homology theory on digraphs, “interesting� digraph must contain a triangle or a square Theorem 2.6 if a digraph has nontrivial a square as sub-digraph.

đ?‘?

(p>2), it must contains a triangle or

2.2 Remark: A categorical view.

Theorem 2.7 (Y.L. Zhang) The formula (2.4) defines a functor from the category of digraphs to the category of abelian groups.

10

ďƒŽ Ap(1) , (ik , ik 1 ) ďƒŽ E (1) , Proof: Let ď š : G1 ď‚Ž G2 be digraph homomorphism, given ei(1) 0 ,...,i p )  eď š(2)(i0 ),...,ď š (ip ) ďƒŽ Ap(2) . Let then (ď Ş (ik ),ď Ş (ik 1 )) ďƒŽ E (2) , for k  0,...n  1, therefore ď š (ei(1) 0 ,...,i p

ď š still denotes the restriction of ď š on ď — p . Therefore ď‚ś(e

p

(1) i0 ,...,i p

p

ď š ď‚ś(ei(1),...,i )  ď š (ďƒĽ (1)q e(1)

and

0

p

q 0

,

i0 ...iq ...i p

p

ďƒ™

i0 ...iq ...i p

q 0

)  ďƒĽ (1) q e(1) ďƒ™

)  ďƒĽ (1) q e(2)

ďƒ™

ď š ( i0 )...ď š ( iq )...ď š (i p )

q 0

. On the other hand,

p

ď š (ei(1),...,i )  eď š(2)(i )...ď š (i ) , and ď‚ś ď š (ei(1),...,i )  ď‚ś(eď š(2)(i )...ď š (i ) )  ďƒĽ (1)q e(2) 0

p

0

notes that e(2)

p

ďƒ™

ď š ( i0 )...ď š ( iq )...ď š ( i p )

0

 e(2)

ďƒ™

p

ď š ( i0 )...ď š ( iq )...ď š ( i p )

0

p

q 0

ďƒ™

ď š ( i0 )...ď š ( iq )...ď š ( i p )

. One

, so we have proved ď š ď‚ś  ď‚ś ď š , i.e. the

diagram

ď —(1) p

ď‚ś

ď š ď —(2) p

ď —(1) p1

ď š ď‚ś

ď —(2) p1

Fig. 2.3 the commutative diagram of boundary operator

and digraph homomorphism

commutes, and ď š : G1 ď‚Ž G2 induced ď š : H p (G1 ) ď‚Ž H p (G2 ) . To conclude, ď ™ is indeed a functor from the category of digraphs to the category of abelian groups.

⎕

3. Laplacians on graphs We shall introduce theories of Laplacians on (undirected) graphs.

The common definition of Laplacians on a connected graph đ??ş ( ) is (đ??ş ) , where D is the incidence matrix of G. Therefore (đ??ş ) has size , (đ??ş ) (đ??ş ), where (đ??ş ) is a and | | An easy observation shows (đ??ş ) diagonal matrix with ii-entry filled by valency of vertice i, A(G) is adjacent matrix of G. One can see such definition in [7] and [8], another definition of Laplacian on directed graphs could be found in [17] and [18], we will also introduce as follow.

We first summarize classical results, and then have a brief view of K.C.Chang’s work of 1-laplacians on graphs.

11

3.1 A survey of classical results I. Basic theorems Here we list some basic results related to and [10].

(đ??ş ), one can find further details in [9]

Lemma 3.1 rk(L(G))=n-1, and the ker (đ??ş ) is the cyclic group generated by the vector ( ) in .

Lemma 3.2 (đ??ş ) is a real symmetric matrix therefore admits n real eigenvalues, with , with and strict positive.

In the following passage, one will see that much of what theories will concentrate on the information determined by the particular eigenvalue .

Lemma 3.3 Let x be a vector, and we have

( x, L(G) x)  xT L(G) x 

ďƒĽ (x

uvďƒŽE

u

2

 xv ) .

II. Potential theory with real coefficient Biggs [10] systematically summarized potential theory on graphs, we only have a look at the first part here, i.e. potential theory with real coefficient.

The theory is based on studying the incidence matrix D, as linear mapping between (đ??ş ) and (đ??ş ), the vector spaces of real-valued functions on V and E, respectively.

Indeed, interpreting the elements of these spaces as column vectors, we have linear operators D : C1 (G; R) ď‚Ž C 0 (G; R), and DT : C 0 (G; R) ď‚Ž C1 (G; R).

Given a

, the formula for

is defined as:

DTď Ą (e)  ď Ą (h(e))  ď Ą (t (e)).

where h means head, and t means tail. for

One first note that Given :

, the defnition of D provides an explicit formula

( Df )(v) 

ďƒĽ

f (e) 

h ( e ) v

Therefore, a function

ďƒĽ

f (e).

t ( e ) v

is in the kernel of D if and only if for all

ďƒĽ

f (e) 

h ( e ) v

ďƒĽ

,

f (e).

t ( e )v

This means that the net accumulation of 12

at every vertice

is zero, and for

this reason a function

is sometimes known as a flow.

âŒŞ ∑ ( ) ( ) According We define inner product for (đ??ş ):⌊ to the general theory of the vector space, there is a orthogonal decomposition of C1 (G, R)  ker D ďƒ… (ker D)ď ž .

Further, [10] part 1 provide another interpretation of the orthogonal decomposition of , as C1 (G, R)  ( flows) ďƒ… (cuts).

where cut is interpreted as characteristic functions of boundary of vertice subset with

,

III. Critical groups Consider (đ??ş ) as an integer matrix as a map from there is a natural short-exact sequence:

to itself. And

i L (G ) ď ° 0 ď‚Ž kerL(G) ď‚žď‚ž ď‚Ž Z n ď‚žď‚žď‚ž ď‚Ž Z n ď‚žď‚ž ď‚Ž cokerL(G) ď‚Ž 0.

(đ??ş ) Since G is connected, The cokernel of (đ??ş ) has the form (đ??ş) (đ??ş) where (đ??ş) is defined to be the critical group of G[11].

It follows from Kirchhoff’s Matrix–Tree Theorem that the order

| (đ??ş)|

(đ??ş),

where (đ??ş) is the number of spanning trees in G. Two integer matrices A and B are equivalent if there exist integer matrices P and Q of determinant such that PAQ = B. Given a square integer matrix A, its Smith normal form is the unique equivalent diagonal matrix ( ) ( ), âˆ? whose entries are nonnegative and divides . The and are known as the invariant factors and determinantal divisors of A respectively.

The structure of the critical group is closely related to the Laplacian matrix: if the Smith normal form of (đ??ş ) is ( ), (đ??ş) is the torsion subgroup of .

Research related to Critical groups and Critical ideals can be found in [12~15].

IV. Cheeger Inequality The famous Cheeger’s inequality from Riemannian Geometry has a discrete analogue involving the Laplacian matrix. This is one of the most important theorems in spectral graph theory. One can find further topics on spectral graph theory in [16].

The Cheeger constant of a graph is a numerical measure of whether or not a graph has a "bottleneck". More formally, the Cheeger constant (đ??ş) of a graph G on n vertices is defined as 13

h(G )  min

S ď‚Łn /2

ď‚ś( S ) .

S

where the minimum is over all nonempty sets S of at most n/2 vertices and ∂(S) is the edge boundary of S, i.e., the set of edges with exactly one endpoint in S. The famous Cheeger Inequality relate the Cheeger constant eigenvalue

ď Ź2 2

(đ??ş) and the second

ď‚Ł h(G) ď‚Ł (2ď Ź2 )1/2 .

(3.1)

3.2 Recent work of 1-laplacian on graphs, by K. C. Chang Traditionally, The Laplace operator is a differential operator acting on functions defined on a manifold M, with ( ). In 2009, K.C.Chang [19] studied nonlinear eigenvalue theory: 1-laplacian operator on , where is defined as (|

), and the eigenvalue problem is to find a pair (

|

)

( ),

satisfying ď „1u ďƒŽ ď ­ sgn(u ) .

Recently(Dec,3,2014), K.C.Chang studied spectrum of 1-laplacian on graphs in [20]. In Chang’s way, the 1−Laplace operator on graphs is defined to be: ď „1x  DT Sgn( Dx).

where D is the incidence matrix, and Sgn: → ( ) is a set valued mapping, with Sgn(y)=(Sgn( ), Sgn( ),‌, Sgn( )), y = ( , ,‌, ), in which

if t > 0 ď&#x192;Ź1 ď&#x192;Ż Sgn(t ) ď&#x20AC;˝ ď&#x192;­ď&#x20AC;­1 if t < 0 ď&#x192;Ż[ď&#x20AC;­1,1] if t = 0 ď&#x192;Ž is a set-valued function.

Main Theorem. (K.C.Chang) Assume that G=(V,E) is connected, then

ď ­2 ď&#x20AC;˝ h(G).

(3.2)

Contrasting (3.1) where Cheeger constant and second eigenvalue are related by an inequality for the linear spectral theory, (3.2) provide an identity between them in 1-Laplacian settings. This is the motivation of Changâ&#x20AC;&#x2122;s study of the nonlinear eigenvalue theory.

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4. Zeta functions It is well known that Zeta function plays an important role in number theory and algebraic geometry. It should be noticed that many papers defined and studied various kinds of zeta functions for undirected graphs, directed graphs or symbolic dynamics.

For example, for a topological dynamic system ( ), where is a topological space, and is a homeomorphism, the Artin-Mazur zeta function for ( ) is defined to be ď&#x201A;Ľ

ď ş T (t ) ď&#x20AC;˝ exp(ď&#x192;Ľ nď&#x20AC;˝1

pn n t ), n

where pn ď&#x20AC;˝ #{x ď&#x192;&#x17D; X : T n x ď&#x20AC;˝ x}.

Kim [32] obtained the Artin-Mazur zeta function for a flip system ( ), i.e, with two homeomorphisms, such that and This could be interpreted as an infinite dihedral group act on . Also, Kim found that when the underlying Z-action is conjugate to a topological Markov shift, the flip system is represented by a pair of matrices, and its zeta function is expressed explicitly in terms of the representation matrices.

For other researches related to this topics, one can refer to [21~31].

5. Graded graphs In this section, we shall introduce graded graphs, differential posets, and related researches, such as Quantized dual graded graphs, Bratteli-Vershik diagram .

5.1 Graded graphs and differential posets A graph is said to be graded if its vertices are divided into levels numbered by integers, so that the endpoints of any edge lie on consecutive levels. Discrete modular lattices and rooted trees are among the typical examples, see Fomin [33].

Definition 5.1 A graded graph is a triple đ??ş

(

), where

(1) V is a discrete set of vertices; (2) (3)

is a rank function, is a multiset of arcs/edges (x,y) where The set

{

( )

that each level is finite and

( )

( )

.

} is called a level of G. Remark: It is always assumed {Ě&#x201A;}

.

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For a graded graph đ??ş, one defines an order on that if and only if a and b lie on consecutive levels and are adjacent, then it could be extended into a partial order. In such way the vertices set becomes a graded poset ( ).

Conversely, a graded poset can induce structure of graded graph. We shall interchangeably use the two terms: graded graph and graded poset.

Differential poset, introduced by R.P. Stanley [34] is a special kind of graded poset, and is a generalization of Young Lattice.

Definition 5.2 A poset P is said to be a differential poset, and in particular to be r-differential (where r is a positive integer), if it satisfies the following conditions: (1) P is graded and locally finite with a unique minimal element Ě&#x201A; ; (2) for every two distinct elements x, y of P, the number of elements covering both x and y is the same as the number of elements covered by both x and y; and (3) for every element x of P, the number of elements covering x is exactly r more than the number of elements covered by x.

As an example of differential poset, we have Proposition 5.3 Youngâ&#x20AC;&#x2122;s Lattice Y is 1-differential. More generally, r-differential.

is

Further, in order to study the combinatorial property of differential poset (especially the enumerative property), two linear transform introduced. This work could be found in R.P. Stanley [34]. We shall briefly list some results here. let K be a field with zero characteristic, and KP be the K-vector space with basis P. Define two linear transform , Ux ď&#x20AC;˝

ď&#x192;Ľ

y,

y covers x

ď&#x192;Ľ

Dx ď&#x20AC;˝

y

x covers y

Theorem 5.4 Let P be a locally finite graded poset with Ě&#x201A; , with finitely many elements of each rank, then P is r-differential poset if and only if Let ( and ( )

(Ě&#x201A;

) denote the number of shortest non-oriented paths between x and y, ) to be the number of paths going from Ě&#x201A; to y. Let

ď Ą (n ď&#x201A;Ž m) ď&#x20AC;˝ ď&#x192;Ľ ď&#x192;Ľ e( x ď&#x201A;Ž y). xď&#x192;&#x17D;Pn yď&#x192;&#x17D;Pm

In other words, ( ) is the number of paths connecting the nth and mth levels.

One can similarily defines ( ). Some enumerative properties are listed as follow: Theorem 5.5 For any graded poset P, one has

ď Ą (0 ď&#x201A;Ž n ď&#x201A;Ž 0) ď&#x20AC;˝ ď&#x192;Ľ e( x)2 . xď&#x192;&#x17D;Pn

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Moreover, if P is r-differential poset, one has

 (0  n  0)   e( x)2  r n n!. xPn

For further researches based on the differential posets, R.P. Stanley [35] identify the

, the poset between level 1 and level 2 of a differential poset P with

hypergraph H. To do this, one first identify the elements of with the integers { } and element with subset { } of { }, where , and is the element in covered by This construction uniquely corresponds

to a hypergraph.

5.2 Quatized dual graded graphs, Bratteli-Vershik diagram Lam [36] studied Quantized dual graded graphs, where the operator relation is Bratteli-Vershik diagram was introduced by Ola, Bratteli[37] in 1972 in the theory of operator algebras to describe directed sequences of finite-dimensional algebras: it played an important role in Elliott's classification of AF-algebras and the theory of subfactors. Subsequently Anatoly Vershik associated dynamical systems with infinite paths in such graphs.[38] As an application, [39] shows that the class of essentially minimal pointed dynamic systems ( ) can be defined and classified in terms of an ordered Bratteli-Vershik diagram.

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