properties of some algebraically defined ......properties of some algebraically defined digraphs...

PROPERTIES OF SOME ALGEBRAICALLYDEFINED DIGRAPHS

Aleksandr Kodess, Felix Lazebnik

Department of Mathematical SciencesUniversity of Delaware

Modern Trends of Algebraic Graph Theory

Aleksandr Kodess, Felix Lazebnik Algebraic Digraphs

What is a digraph?

v5DefinitionA digraph is a pair D = (V ,A)of:

a set V , whose elementsare called vertices ornodesa set A of ordered pairs ofvertices, called arcs,directed edges, or arrows

What is an algebraic digraph D(q; f )?

LetFq be a finite field with q elements;f : F2

q → Fq be a bivariate polynomial.

DefinitionAn algebraic digraph, denoted D(q; f ), is a digraph whose

vertex set is F2q

arc set consits of ordered pairs(

(x1, x2), (y1, y2))

with therelation

x2 + y2 = f (x1, y1),

vertex set is F2q

(x1, x2), (y1, y2))

with therelation

x2 + y2 = f (x1, y1),

vertex set is F2q

(x1, x2), (y1, y2))

with therelation

x2 + y2 = f (x1, y1),

vertex set is F2q

(x1, x2), (y1, y2))

with therelation

x2 + y2 = f (x1, y1),

vertex set is F2q

(x1, x2), (y1, y2))

with therelation

x2 + y2 = f (x1, y1),

Example of D(q; f )

V (D) = F2q

f : F2q → Fq

There is an arc from vertex (x1, x2) to vertex (y1, y2) if andonly if

x2 + y2 = x2 + xy + y2 + 1 = f1(x1, y1)x2 + y2 = x2 + xy + y2 = f2(x1, y1)x2 + y2 = xy = f3(x1, y1)

Easy to argue that if q is odd, then

D(q; f1) ∼= D(q; f2) ∼= D(q; f3).

Example of D(q; f )

V (D) = F2q

f : F2q → Fq

D(q; f1) ∼= D(q; f2) ∼= D(q; f3).

Example of D(q; f )

V (D) = F2q

f : F2q → Fq

D(q; f1) ∼= D(q; f2) ∼= D(q; f3).

Example of D(q; f )

V (D) = F2q

f : F2q → Fq

D(q; f1) ∼= D(q; f2) ∼= D(q; f3).

Simple Observation

Simple isomorphisms

Let q be an odd prime power, and f ∈ Fq[x , y ]. Letf1(x , y) = f (x , y)− f (0,0), andf ∗(x , y) = f1(x , y)− f (x ,0)− f (0, y). The following statementshold:

D(q; f ) ∼= D(q; f1).If, in addition, f is a symmetric polynomial, then

D(q; f ) ∼= D(q; f1) ∼= D(q; f ∗).

Simple Observation

Simple isomorphisms

D(q; f ) ∼= D(q; f1) ∼= D(q; f ∗).

Simple Observation

Simple isomorphisms

D(q; f ) ∼= D(q; f1) ∼= D(q; f ∗).

What is a monomial algebraic digraph?

DefinitionA monomial algebraic digraph, denoted D(q; m,n), is analgebraic digraph in which

vertex set V is F2q

there is an arc from (x1, x2) to (y1, y2) if and only if

x2 + y2 = xm1 yn

D(3; 1,2)

Motivation

Work of:Lazebnik, Woldar (2001)Lazebnik, Ustimenko (1993, 1995, 1996)Viglione (2001)Dmytrenko, Lazebnik, Viglione (2005)

Bipartite undirected graph BΓn

V (BΓn) = Pn ∪ Ln, both Pn and Ln are copies of Fnq

point (p) = (p1, . . . , pn) is adjacent to line [l] = (l1, . . . , ln) if

l2 + p2 = f2(p1, l1)l3 + p3 = f3(p1, l1,p2, l2)

...ln + pn = fn(p1, l1,p2, l2, . . . , pn−1, ln−1).

Properties of BΓn

BΓn admits neighbor-complete coloring, i.e. every color isuniquely represented among the neighbors of each vertexcovering properties of BΓn. For instance, BΓn covers BΓkfor n > k .embedded spectra properties. For instance,spec(BΓk ) ⊆ spec(BΓn) for k < n.edge-decompostiion properties. We have BΓndecomposing Kqn,qn .

Properties of BΓn

An application of BΓn with a certain specialization

LetCn denote the cycle of length n ≥ 3ex(v , {C3,C4, . . . ,C2k}) denote the greatest number ofedges in a graph or order v which contains no subgraphsisomorphic to any C3, . . . ,C2k .

Theorem (Lazebnik, Ustimenko, Woldar 1995)

ex(v , {C3,C4, . . . ,C2k}) ≥ ckv1+ 23k−3+ε ,

where ck is a positive function if k, and ε = 0 if k 6= 5 is odd,and ε = 1 if k is even.

This lower bounds comes from BΓn with a certain choice ofdefining functions fi , 2 ≤ i ≤ n.

An application of BΓn with a certain specialization

LetCn denote the cycle of length n ≥ 3ex(v , {C3,C4, . . . ,C2k}) denote the greatest number ofedges in a graph or order v which contains no subgraphsisomorphic to any C3, . . . ,C2k .

Theorem (Lazebnik, Ustimenko, Woldar 1995)

ex(v , {C3,C4, . . . ,C2k}) ≥ ckv1+ 23k−3+ε ,

where ck is a positive function if k, and ε = 0 if k 6= 5 is odd,and ε = 1 if k is even.

This lower bounds comes from BΓn with a certain choice ofdefining functions fi , 2 ≤ i ≤ n.

Some Upper Bounds on ex(v , {C3,C4, . . . ,C2k})

Erdos, Bondy–Simonovits, 1974

ex(v , {C3,C4, . . . ,C2k}) ≤ 20kv1+(1/k) , for v sufficiently large.

Verstraëte, 2000

ex(v , {C3,C4, . . . ,C2k}) ≤ 8(k − 1)v1+(1/k), for v sufficientlylarge.

Pikhurko, 2012

ex(v , {C3,C4, . . . ,C2k}) ≤ (k − 1)v1+(1/k) + O(v) .

Bukh, Jiang, 2014

ex(v , {C3,C4, . . . ,C2k}) ≤ 80√

k log k · v1+(1/k) + O(v) .

Verstraëte, 2000

Pikhurko, 2012

ex(v , {C3,C4, . . . ,C2k}) ≤ (k − 1)v1+(1/k) + O(v) .

Bukh, Jiang, 2014

ex(v , {C3,C4, . . . ,C2k}) ≤ 80√

k log k · v1+(1/k) + O(v) .

Verstraëte, 2000

Pikhurko, 2012

ex(v , {C3,C4, . . . ,C2k}) ≤ (k − 1)v1+(1/k) + O(v) .

Bukh, Jiang, 2014

ex(v , {C3,C4, . . . ,C2k}) ≤ 80√

k log k · v1+(1/k) + O(v) .

Verstraëte, 2000

Pikhurko, 2012

ex(v , {C3,C4, . . . ,C2k}) ≤ (k − 1)v1+(1/k) + O(v) .

Bukh, Jiang, 2014

ex(v , {C3,C4, . . . ,C2k}) ≤ 80√

k log k · v1+(1/k) + O(v) .

Questions studied

Strong Connectivity of D(q; f )

Classification into Isomorphism Classes

Questions studied

Strong Connectivity of D(q; f )

Classification into Isomorphism Classes

Strong Connectivity

DefinitionA directed graph is calledstrongly connected if it containsa directed path from u to vand a directed path from v tou for every pair of distinctvertices u, v .

D(3; 1,2) revisited

(Strong) Connectivity of Algebraic Digraphs

Definition (Linked alternating sums)

For any function f : F2q → Fq, any positive integer k ≥ 2 and any

k -tuple (x1, . . . , xk ) ∈ Fkq we call the sum

f (x1, x2)− f (x2, x3) + · · ·+ (−1)k−1f (xk−1, xk )

a linked alternating sum of f of length k − 1. Let LSk (f ) be theset of all possible linked alternating sums of f of length k − 1,and let

LS(f ) :=⋃k≥2

LSk (f ).

We call LS(f ) the set of linked alternating sums of f .

Some additional notation

Im(f ) — image of f : F2q → Fq

Im(f ) := {(f (x , y), : x , y ∈ Fq}.

Affine subspaces H0 and H1

For any vectors α1, . . . , αd ∈ Fq, let H0 and H1 be defined as

H0 = H0(α1, . . . , αd ) =

siαi : all si in Fp,

d∑i=1

si = 0

H1 = H1(α1, . . . , αd ) =

siαi : all si in Fp,

d∑i=1

si = 1

Then H0 is a subspace of Fq, and H1 is an affine subspace ofFq.

Property of the set LS(f)

Trivially, LS(f ) ⊆ 〈Im(f )〉

EitherLS(f ) = 〈Im(f )〉, span of image of f in Fq, OR

LS(f ) = H0 ∪ H1, union of two affine planes in Fq

This leads to the following

Strong Connectivity of D(p; f )

TheoremLet p be an odd prime, and let f : Fp × Fp → Fp. The followingstatements hold.

(i) If f is not a symmetric function, i.e. there exist x1, x2 ∈ Fpfor which f (x1, x2) 6= f (x2, x1), then D(p; f ) is strong.

(ii) If f is a symmetric function, let

f ∗(x , y) = f (x , y)− f (x ,0)− f (0, y)− f (0,0).

If f ∗ is a nonzero polynomial, then D(p; f ) is strong. If f ∗ isthe zero polynomial, then D(p; f ) has (p + 1)/2 strongcomponents. The component containing the vertex (0,0) isisomorphic to the complete looped digraph

−→K p. All other

strong components are isomorphic to the completebipartite digraph

−→K p,p with no loops.

Isomorphism Classification Problem

ProblemWhat are necessary and sufficient conditions for (m1,n1) and(m2,n2) in order for the graphs D(q; m1,n1) and D(q; m2,n2) tobe isomorphic?

Idea to Solve it — find a strong easily computable digraphinvariant

number of cyclesnumber of ’triangles’ with various orientations of arcsnumber of subdigraphs on 4 verticesnumber of ’looped’ paths

Dmytrenko, Lazebnik,Viglione, 2005 — undirected bipartiteversionThis worked for undirected bipartite monomial graphs. Theyfound the following simple graph invariants:

number of C4 for all sufficiently large q’snumber of Ks,t for all q’s.as multisets {m1,n1} = {m2,n2}, where x = gcd(q − 1, x)

Directed versionWe have examples of non-isomorphic graphs that have equalnumbers of 3,4,5,6-cycles and a large collection of othersubgraphs.

Example: Digraphs D(169; 1,5) and D(169; 1,101) have equalnumber of directed cycles Ck , for 3 ≤ k ≤ 13, but they are notisomorphic.

Open Conjecture

Conjecture

Let q be a prime power, 1 ≤ m,n,m′,n′ ≤ q − 1 integers. ThenD(q; m1,n1) ∼= D(q; m2,n2) if and only if there exists k coprimewith q − 1 such that

m2 ≡ km1 mod q − 1,

n2 ≡ kn1 mod q − 1.

Commentseasy to prove sufficiencycannot prove necessitychecked with sage for orders of up to 1000

Open Conjecture

Conjecture

m2 ≡ km1 mod q − 1,

n2 ≡ kn1 mod q − 1.

Open Conjecture

Conjecture

m2 ≡ km1 mod q − 1,

n2 ≡ kn1 mod q − 1.

Open Conjecture

Conjecture

m2 ≡ km1 mod q − 1,

n2 ≡ kn1 mod q − 1.

Some comments on Conjecture

Using reduction to bipartites graphs

D(q; m1,n1) ∼= D(q; m2,n2) implies

{m1,n1} = {m2,n2}

If, say, m1 = m2 and n1 = n2, then

m2 ≡ k1m1 mod q − 1,

n2 ≡ k2n1 mod q − 1.

We cannot conclude k1 = k2. However, we have noexamples of isomorphic digraphs with k1 6= k2.

{m1,n1} = {m2,n2}

m2 ≡ k1m1 mod q − 1,

n2 ≡ k2n1 mod q − 1.

{m1,n1} = {m2,n2}

m2 ≡ k1m1 mod q − 1,

n2 ≡ k2n1 mod q − 1.

Conjecture proved under restrictions for p > 2

TheoremLet q = p > 2 be prime and 1 ≤ m1,n1,m2,n2 ≤ p − 1 beintegers. Suppose that D1 = D(p; m1,n1) ∼= D2 = D(p; m2,n2).Assume moreover that there exists an isomorphismφ : V (D1)→ V (D2) of the form

φ : (x , y) 7→ (f (x , y),g(x , y)),

in which f depends on x only. If m1 6= n1 or m2 6= n2, then thereexists an integer k, coprime with p − 1, such that

m2 ≡ km1 mod p − 1,

n2 ≡ kn1 mod p − 1.

properties of some algebraically defined ......properties of some algebraically defined digraphs...

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