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 sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Outline Counting Closed Walks 1 Extensions for 4-Regular Bipartite Graphs 2 Extensions for Regular Graphs 3 sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
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 P A ( 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 , 1 2 , 0 , − 1 , − 2 2 } sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Counting Closed Walks � λ ℓ w ℓ = i = # closed walks of length ℓ in G i w 0 = n w 1 = 0 w 2 = 2 m w 3 = 6 [ C 3 ] w 4 � = x [ C 4 ] [ H ] -> # of (not necessarily induced) subgraphs of G sciLogo.png isomorphic to H Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Extensions of w ℓ [Boulet and Jouve 2008] A lollipop graph L ( m , k ) is the coalescence of C m and P k + 1 (at an endpoint). An ℓ - covering closed walk in H , ω ℓ ( H ) , is a closed walk of length ℓ running through all edges of H at least once. � w ℓ = ω ℓ ( H )[ H ] H ∈{ H | ω ℓ ( H ) > 0 } Ex// Given a graph with no C 3 and no C 5 subgraphs, w 6 = 12 [ C 6 ]+ 2 [ P 2 ]+ 12 [ P 3 ]+ 6 [ P 4 ]+ 12 [ K 1 , 3 ]+ 48 [ C 4 ]+ 12 [ L ( 4 , 1 )] . sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Covering Closed Walk w ℓ 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 w ℓ extensions: Boulet and Jouve 2008 Lollipops L ( m , k ) are DS. sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs More Walk Definitions Consider a walk W = v 0 v 1 · · · v ℓ . If v i − 1 = v i + 1 for some i then W is reducible (otherwise W is irreducible ). (If reducible) W can be reduced at index i by omitting v i and v i + 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 . sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Extensions of w ℓ [Cvetkovic 1998, Stevanovic 2007] W i = v 0 v 1 · · · v i is a prefix of W , 0 ≤ i ≤ ℓ . If red ( W i ) 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 w 4 = 28 n + 8 [ C 4 ] w 6 = 232 n + 144 [ C 4 ] + 12 [ C 6 ] w 8 ≥ 2092 n + 2024 [ C 4 ] + 288 [ C 6 ] sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
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// C 3 , C 4 , C 6 , K n , P 2 , 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. sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
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 , 3 x , 2 y , 1 z , 0 2 w , − 1 z , − 2 y , − 1 x , − 4 } n x y z w C 4 C 6 8 0 0 0 3 36 96 10 0 0 4 0 30 130 12 0 1 4 0 27 138 12 0 2 0 3 30 112 14 1 0 3 2 36 102 . . . 560 76 84 84 35 0 0 The first such list [Cvetkovic, Simic, Stevanovic 1998] The improved list [Stevanovic et al. 2007] 43 different values for n sciLogo.png 828 different entries Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs List Building Tools Diophantine equations (inequalities) for w ℓ , 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 k � � ( r − µ i ) J = n ( A − µ i I ) i = 2 i = 2 where µ 1 , µ 2 , ..., µ k are the distinct eigenvalues, J is the all 1s matrix, and I is the identity matrix Graph angles equations ( α ij : the angles of G ) k � α 2 ij µ ℓ i = # closed walks of length ℓ from vertex j sciLogo.png i = 1 Marsha Kleinbauer Counting walks and the resulting polynomials
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 one of the possible spectra are found, listed and drawn [Stevanovic et al. 2007]. Figure : Sp ( G ) = { 4 , 0 6 , − 4 } sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs McKay’s Result Let u ℓ be the number of totally-reducible walks of length ℓ in G . McKay 1981 Let G be an r -regular graph. For even ℓ , ℓ/ 2 � ℓ � ℓ − 2 i + 1 � ℓ − i + 1 r i u ℓ = n i i = 0 A totally-reducible walk must have even length, so u ℓ = 0 for all odd ℓ . sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Extending w ℓ [K. and Wanless] The idea for regular G : Use u ℓ to count totally-reducible walks. Count not totally-reducible walks z ℓ : Use a generating function to count closed walks containing a cycle C k . Use a generating function to count closed walks containing a polycyclic subgraph w ℓ = u ℓ + z ℓ We count z ℓ by extending walks from a set of base walks . sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Extending Walks To W = v 0 v 1 ... v ℓ we add the following extras: A diversion : a closed walk v i d i of length ≥ 0 occuring in the place of v i for some 0 < i ≤ ℓ such that red ( v i d i ) = v i and no intermediate step of the reduction results in v i v i + 1 ... v i Result: W ′ = v 0 v 1 ... v i d i v i + 1 ... v ℓ A tail : a pair of walks u 1 u 2 ... u t v 0 and v ℓ u t ... u 1 where t ≥ 1 occuring in the place of v 0 and v ℓ respectively with u 1 u 2 ... u t v 0 irreducible, u t � = v 1 , and u t � = v ℓ − 1 Result: W ′ = u 1 ... u t v 0 v 1 ... v ℓ u t ... u 1 sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs The Base Walks W = v 0 v 1 ... v ℓ is a Base Walk if ℓ > 0 W is closed W is irreducible W has no tail ex// 0120 − − > 30141203 sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
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 k 1 and every other vertex has degree k 2 + 1 . The generating function for closed rooted walks in T is 2 k 2 T k 1 = � 1 − 4 x 2 k 2 2 k 2 − k 1 + k 1 This result: Wanless 2010 , Similar results: Quenell 1994 , Chung and Yau 1999 sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
Counting Closed Walks Extensions for 4-Regular Bipartite Graphs Extensions for Regular Graphs Generating Functions for z ℓ Let G be an ( r + 1 ) -regular graph. Wanless 2010 Let W = v 0 v 1 ... v k be a walk of length k in G . The generating function for walks in G that are formed by adding diversions to W is x k T k r T r + 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 � 1 − x 2 T 2 r ψ ( l ) = x k T k � r T r + 1 1 − rx 2 T 2 r sciLogo.png Marsha Kleinbauer Counting walks and the resulting polynomials
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