Counting walks and the resulting polynomials Marsha Kleinbauer TU - - PowerPoint PPT Presentation

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Counting walks and the resulting polynomials

Marsha Kleinbauer

TU Kaiserslautern, Germany

Graph Polynomials: Towards a Comparative Theory, 2016

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Outline

1

Counting Closed Walks

2

Extensions for 4-Regular Bipartite Graphs

3

Extensions for Regular Graphs

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Initial Items

G is a simple and connected graph with adjacency matrix A. The roots of the characteristic polynomial PA(G; x) = det(xI − A) are the eigenvalues of G. The spectrum of a graph, Sp(G), is the set of eigenvalues with their multiplicity. Sp(G) = {3, 12, 0, −1, −22}

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Counting Closed Walks

wℓ =

• i

λℓ

i = # closed walks of length ℓ in G

w0 = n w1 = 0 w2 = 2m w3 = 6[C3] w4 = x[C4] [H] -> # of (not necessarily induced) subgraphs of G isomorphic to H

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Extensions of wℓ [Boulet and Jouve 2008]

A lollipop graph L(m, k) is the coalescence of Cm and Pk+1 (at an endpoint). An ℓ-covering closed walk in H, ωℓ(H), is a closed walk of length ℓ running through all edges of H at least

• nce.

wℓ =

• H∈{H|ωℓ(H)>0}

ωℓ(H)[H] Ex// Given a graph with no C3 and no C5 subgraphs, w6 = 12[C6]+2[P2]+12[P3]+6[P4]+12[K1,3]+48[C4]+12[L(4, 1)].

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Covering Closed Walk wℓ Results

Motivation #1: Which graphs are determined by their spectrum (DS)? Haemers, Liu, and Zhang 2008 Lollipops L(m, k) with m odd are DS. Using a combination of their wℓ extensions: Boulet and Jouve 2008 Lollipops L(m, k) are DS.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

More Walk Definitions

Consider a walk W = v0v1 · · · vℓ. If vi−1 = vi+1 for some i then W is reducible (otherwise W is irreducible). (If reducible) W can be reduced at index i by omitting vi and vi+1. red(W) is the irreducible result of repeatedly reducing at some i. red(W) is unique [Godsil 1981]. If red(W) is trivial then we say that W is totally-reducible.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Extensions of wℓ [Cvetkovic 1998, Stevanovic 2007]

Wi = v0v1 · · · vi is a prefix of W, 0 ≤ i ≤ ℓ. If red(Wi) is a path for each i then W is called a tree-like walk. A tree-like walk is closed if and only if it is totally-reducible. The idea: prove that counting closed walks in an r-regular graph is equivalent to counting them in an r-regular infinite tree use a recurrence relation to count all closed walks around cycles specifically for r = 4 Given a 4-regular bipartite graph w4 = 28n + 8[C4] w6 = 232n + 144[C4] + 12[C6] w8 ≥ 2092n + 2024[C4] + 288[C6]

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Motivation # 2: Which graphs have integral spectra? [Harary and Schwenk 1974] An integral graph is a graph whose eigenvalues are integers. Ex// C3, C4, C6, Kn, P2, cube, triangular prism Connected integral graphs with n vertices n 1 2 3 4 5 6 7 8 9 10 11 12 # 1 1 1 2 3 6 7 22 24 83 113 325? Bussemaker and Cvetkovic 1976, Schwenk 1978 There are exactly 13 connected cubic integral graphs.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Lists of possible spectra

4-Regular Bipartite Integral Graphs: Sp(G) = {4, 3x, 2y, 1z, 02w, −1z, −2y, −1x, −4}

n x y z w C4 C6 8 3 36 96 10 4 30 130 12 1 4 27 138 12 2 3 30 112 14 1 3 2 36 102 . . . 560 76 84 84 35

The first such list [Cvetkovic, Simic, Stevanovic 1998] The improved list [Stevanovic et al. 2007]

43 different values for n 828 different entries

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

List Building Tools

Diophantine equations (inequalities) for wℓ, ℓ = 0, 2, 4, 6, 8 Upper bound on the number of vertices of G (radius R): n ≤ 2(r − 1)R − 2 r − 2 A Lemma of Hoffman:

k

• i=2

(r − µi)J = n

k

• i=2

(A − µiI) where µ1, µ2, ..., µk are the distinct eigenvalues, J is the all 1s matrix, and I is the identity matrix Graph angles equations (αij: the angles of G)

k

• i=1

α2

ijµℓ i = # closed walks of length ℓ from vertex j

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

New Found Integral Graphs

All 4-Regular Bipartite Integral Graphs with n ≤ 24 that realize

• ne of the possible spectra are found, listed and drawn

[Stevanovic et al. 2007].

Figure : Sp(G) = {4, 06, −4}

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

McKay’s Result

Let uℓ be the number of totally-reducible walks of length ℓ in G. McKay 1981 Let G be an r-regular graph. For even ℓ, uℓ = n

ℓ/2

• i=0

ℓ i ℓ − 2i + 1 ℓ − i + 1 r i A totally-reducible walk must have even length, so uℓ = 0 for all

• dd ℓ.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Extending wℓ [K. and Wanless]

The idea for regular G: Use uℓ to count totally-reducible walks. Count not totally-reducible walks zℓ:

Use a generating function to count closed walks containing a cycle Ck. Use a generating function to count closed walks containing a polycyclic subgraph

wℓ = uℓ + zℓ We count zℓ by extending walks from a set of base walks.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Extending Walks

To W = v0v1...vℓ we add the following extras: A diversion: a closed walk vidi of length ≥ 0 occuring in the place of vi for some 0 < i ≤ ℓ such that red(vidi) = vi and no intermediate step of the reduction results in vivi+1...vi Result: W ′ = v0v1...vidivi+1...vℓ A tail: a pair of walks u1u2...utv0 and vℓut...u1 where t ≥ 1 occuring in the place of v0 and vℓ respectively with u1u2...utv0 irreducible, ut = v1, and ut = vℓ−1 Result: W ′ = u1...utv0v1...vℓut...u1

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

The Base Walks

W = v0v1...vℓ is a Base Walk if ℓ > 0 W is closed W is irreducible W has no tail ex// 0120 − − > 30141203

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Tree-Like Walk Generating Function

Lemma Let T be an infinite rooted tree in which the root has degree k1 and every other vertex has degree k2 + 1. The generating function for closed rooted walks in T is Tk1 = 2k2 2k2 − k1 + k1

• 1 − 4x2k2

This result: Wanless 2010, Similar results: Quenell 1994, Chung and Yau 1999

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Generating Functions for zℓ

Let G be an (r + 1)-regular graph. Wanless 2010 Let W = v0v1...vk be a walk of length k in G. The generating function for walks in G that are formed by adding diversions to W is xkT k

r Tr+1.

• K. and Wanless

Suppose W is a closed walk in G of length k. The generating function for walks in G that are extensions of W is ψ(l) = xkT k

r Tr+1

1 − x2T 2

r

1 − rx2T 2

r

• Marsha Kleinbauer

Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Extending wℓ [K. and Wanless]

For each subgraph H, we generate all walks, ν(H; x) containing H including the possible extensions.

• i≤ℓ zixi =

H∈Hℓ ν(H; x)[H] + O(xℓ+1).

Ex// For an (r + 1)-regular graph, w4 = (1 + 3r + 2r 2)n + 8[C4], w5 = 30r[C3] + 10[C5], w6 = (1 + 5r + 9r 2 + 5r 3)n + 6[C3] + 48r[C4] + 12[C6] + 24[C3·3] + 12[Θ2,2,1], w7 = 126r 2[C3] + 70r[C5] + 14[C7] + 28[C4·3] + 28[Θ2,2,1] + 14[Θ3,2,1] + 84[Θ2,2,2,1].

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Counting Cycles at Each Vertex

#Ck per vertex = k ∗ (#Ck) n ∈ Z+ Ex// Given that n = 48, Sp(G) = {4, 35, 26, 111, 02, −111, −26, −35, −4}, and that C4 = 24 and C6 = 140; k ∗ (#Ck) n = 4 ∗ (24) 48 = 2 ∈ Z+ k ∗ (#Ck) n = 6 ∗ (140) 48 = 35 2 / ∈ Z+ ∴ The possible spectra list entry, 48 5 6 11 1 24 140, can’t be realized by a vertex-transitive graph.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Using wℓ Equations

The # of closed walks of length 8 in a bipartite graph G:

n

• i=1

λ8

i =2092n + 2024[C4] + 288[C6] + 16[C8] + 96[Θ2,2,2,2]

+ 48[Θ2,2,2] + 16[Θ3,3,1] where Θi1,i2,...,ih consists of two vertices joined by internally disjoint paths of lengths ij for j = 1, . . . , h. Often [C8] could now be determined and then 8∗[C8]

n

was used to eliminate possible spectra unable to realize a vertex-transitive graph.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Possible Spectra List for the Vertex Transitive Case

List of possible spectra - connected 4-regular integral bipartite: 8 ≤ n ≤ 560 43 different values for n 828 different entries List of possible spectra - connected 4-regular vertex-transitive integral bipartite: 8 ≤ n ≤ 560 29 different values for n 58 different entries

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Results on 4-Regular Integral G [K. and Wanless]

There are exactly 32 4-Regular Integral Cayley Graphs. There are exactly 27 4-Regular Integral Graphs that are arc-transitive.

The Only Cayley Integral Bipartite Graphs n 8 10 12 16 18 24 30 32 36 40 48 72 120 # 1 1 2 1 1 3 1 1 1 1 1 2 1 Sp(G) = {4, 212, 06, −212, −4}

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Summary

The following questions are still largely unanswered: Which graphs are determined by their spectrum? Which graphs have integral spectra? The smallest entry of the 4-regular integral bipartite graphs possible spectra list, for which the existence question is not completely solved has 28 vertices. The smallest entry of the corresponding vertex-transitive list, for which the existence question is not completely solved has 36 vertices.

Marsha Kleinbauer Counting walks and the resulting polynomials

sciLogo.png Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs

Thanks!

Marsha Kleinbauer Counting walks and the resulting polynomials