W HAT DOES IT COUNT ? If G is a graph on n vertices then the - - PowerPoint PPT Presentation

THE CHARACTERISTIC POLYNOMIAL OF A GRAPH

Gordon Royle

Centre for the Mathematics of Symmetry & Computation School of Mathematics & Statistics University of Western Australia

Dagstuhl, June 2016

AUSTRALIA

PERTH

UNIVERSITY OF WESTERN AUSTRALIA

OUTLINE

INTRODUCTION SOME BASICS RECONSTRUCTION DS GRAPHS

GRAPHS

The Petersen graph is a

I cubic graph, I with 10 vertices, I and 15 edges.

CHARACTERISTIC POLYNOMIAL

If the graph G has 0/1-adjacency matrix A, then the characteristic polynomial of G is ϕ(G; x) = det(xI A). It is the usual characteristic polynomial from linear algebra (not the characteristic polynomial of matroid theory).

EXAMPLE

A(P) =                1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1               

ϕ(P; x) = det(xI A(P)) = (x 3)(x 1)5(x + 2)4

THE SPECTRUM

As A(G) is symmetric, the roots of ϕ(G), i.e., the eigenvalues of A(G), are all real. This collection of real roots of A(G) is called the spectrum of G: spec(P) = {3, 15, 24}.

SPECTRAL GRAPH THEORY

Spectral graph theory is the study of the relationship between graphical properties and graph spectra.

SOME THEMES

I Second-largest eigenvalue

Expansion properties and expanders

I Smallest eigenvalue

Independence and (hence) colouring properties

I Interlacing

Subgraphs and other substructures

I Computable invariant

Isomorphism and DS questions

I Few eigenvalues

Highly-structured graphs

I Integral multiplicities

Existence and non-existence

I Sum of eigenvalues

Energy of (molecular) graphs

OUTLINE

INTRODUCTION SOME BASICS RECONSTRUCTION DS GRAPHS

WHAT DOES IT COUNT?

If G is a graph on n vertices then the coefficient of xn−r in ϕ(G; x) is X

H

(1)comp(H)2cyc(H) where H ranges over all r-vertex subgraphs of G such that

I H consists of disjoint edges and cycles, I comp(H) is the number of components of H, I cyc(H) is the number of cycles in H.

WHY?

We know that for any n ⇥ n matrix N, det(N) = X

σ∈Sym(n)

sgn(σ) Y

v

Nvσ(v) In order for σ to contribute, the product must be non-zero so Nvσ(v) 6= 0 for all v 2 {1, . . . , n}. Applying this to xI A shows that {vσ(v) | v 6= σ(v)} is a set of edges of G inducing a subgraph consisting of disjoint cycles and single edges.

σ = (1,3,5,2,4)(6,7)(8)(9,10)

(1, 3, 5, 2, 4)(6, 7)(8)(9, 10)

                     x −1 −1 −1 x −1 −1 −1 −1 x −1 −1 −1 −1 x −1 −1 −1 −1 −1 x −1 −1 x −1 −1 −1 −1 x −1 −1 −1 x −1 −1 −1 x −1 −1 −1 −1 x                     

IMMEDIATE CONSEQUENCES

I If G has no cycles

Equivalent to counting matchings — matching polynomial

I If G has no odd cycles

All non-zero contributions are to xn−even so ϕ(G; x) = p(x2) or x p(x2) for some polynomial p, so the spectrum is symmetric about 0.

FUNDAMENTAL QUESTIONS

For any graph polynomial P, two (closely related) fundamental questions are:

I What graphical properties can be determined from P? I What graphs are determined (up to isomorphism) by P?

COSPECTRALITY

As with all graph polynomials, the existence of non-isomorphic graphs G, H such that ϕ(G) = ϕ(H) provides information about graphical properties not determined by ϕ. Graphs with the same characteristic polynomial are called cospectral.

THE SALTIRE PAIR

ϕ(x) = x3(x2 4)

THE SALTIRE PAIR

ϕ(x) = x3(x2 4)

The Saltire

WHAT IS not DETERMINED

From the Saltire pair alone:

I Whether graph is connected I The degree sequence of a graph I The girth of a graph

WHAT is DETERMINED?

The spectrum determines:

I The number of vertices and edges I The number of triangles I The number of closed walks of every length I If the graph is bipartite

WALKS

These results follow from a few simple facts:

I The vw-entry of A2 counts the 2-step walks from v to w. I The vw-entry of Ak counts the k-step walks from v to w. I The total number of closed walks of length k is tr Ak. I The trace of Ak is P λ λk.

AND REGULARITY

The regularity of a graph can be determined from its spectrum: A graph is k-regular if and only if λ1 = k and X

λ

λ2 = kn. This follows because for a non-regular graph kmin < ¯ k < λ1 < kmax.

OUTLINE

INTRODUCTION SOME BASICS RECONSTRUCTION DS GRAPHS

THE DECK OF A GRAPH

The deck of G is the multiset of (unlabelled) vertex-deleted subgraphs deck(G) = {G\v : v 2 V(G)} Which graph(s) G have this particular deck?

THE RECONSTRUCTION CONJECTURE

Ulam’s reconstruction conjecture is that non-isomorphic graphs1 have different decks: deck(G) = deck(H) ) G ' H. A graphical parameter is reconstructible if its value can be determined from the deck.

1on at least 3 vertices

CHARACTERISTIC POLYNOMIAL

Elementary linear algebra shows us that d dx (ϕ(G; x)) = X

v∈V(G)

ϕ(G\v; x) So all but one of the coefficients of ϕ(G) are (immediately) determined by the deck. The “missing coefficient” is the constant term of ϕ(G), namely — the determinant of A(G).

TUTTE

In his famous paper “All the King’s Horses”, Bill Tutte showed that a variety of graph parameters are reconstructible.

RECONSTRUCTIBLE PARAMETERS

I The Tutte polynomial, hence chromatic, flow polynomials I The number of spanning trees I The number of hamilton cycles I The determinant of A(G)

Therefore ϕ is reconstructible. (So counterexamples to the reconstruction conjecture have to be cospectral, coTutte, . . .)

THE POLYNOMIAL DECK

Suppose we replace the deck of vertex-deleted subgraphs with the polynomial deck: polynomial deck(G) = {ϕ(G\v) : v 2 V(G)} x4 3x2 x4 3x2 x4 3x2 x4 3x2 x4 Can the characteristic polynomial of G be reconstructed from the polynomial deck of G? This is an open problem.

TWO PARTIAL RESULTS

THEOREM (HAGOS 2000)

The characteristic polynomial of a graph G is reconstructible from the set of ordered pairs {(ϕ(G\v), ϕ(G\v) | v 2 V} where G is the complement of G.

THEOREM (CVETKOVI ´

C AND LEPOVI ´ C 1998)

The characteristic polynomial of a tree T is reconstructible from the polynomial deck of T.

OUTLINE

INTRODUCTION SOME BASICS RECONSTRUCTION DS GRAPHS

DS GRAPHS

A graph is called DS if it is Determined by its Spectrum — in other words, no other graph has the same characteristic polynomial. It turns out to be easy to find cospectral graphs.

SMALL GRAPHS

Even with extra conditions (in this case, regularity) there are many cospectral small graphs.

The smallest connected cubic cospectral graphs

STRONGLY REGULAR GRAPHS

A graph is strongly regular with parameters (n, k, λ, µ) if

I It has n vertices, I It is k-regular I Adjacent vertices have λ common neighbours I Distinct non-adjacent vertices have µ common neighbours

All SRGs with the same parameters have the same spectrum. There are bucketloads2 of strongly regular graphs — for example, 11 billion 57-vertex SRGs from Steiner Triple Systems with 19 points.

2a technical term

GODSIL/MCKAY SWITCHING

Ingredients (simplified version):

I A symmetric b ⇥ b matrix B with constant row sums I A symmetric c ⇥ c matrix C I A b ⇥ c matrix N with all column sums in {0, b/2, b}

GODSIL/MCKAY SWITCHING

Then form two matrices A1 =  B N NT C

• A2 =

 B b N b NT C

• where J is an all-ones matrix and b

N is obtained from N by exchanging

• nes and zeros in all the columns of weight b/2.

Then A1 and A2 are similar matrices.

PROOF

Let Q =  2

bJb Ib

Ic

• Then Q2 = Ib+c and

QA1Q = A2

GRAPHICALLY

Graphically this gives a partition with the “none, half or all” property. and cospectral graphs are obtained by switching on the b/2-vertices.

EXAMPLE

These both have

ϕ(x) = (x + 1)2(x8 − 2x7 − 16x6 + 16x5 + 72x4 − 42x3 − 96x2 + 44x + 7)

But are they isomorphic? (no)

SCHWENK’S FAMOUS RESULT

In the 1970s, Allen Schwenk proved the following famous result:

THEOREM

Almost all trees are cospectral. If T is the tree then it is routine to check that ϕ(T\ ) = ϕ(T\ )

SCHWENK’S PROOF

Two facts, one easy, one not:

I Coalescing two trees S, T at a vertex v gives a tree with

ϕ(S v T) = ϕ(S)ϕ(T\v) + ϕ(S\v)ϕ(T) xϕ(S\v)ϕ(T\v)

I Almost all trees contain the 9-vertex tree from the previous slide

DO WE KNOW any (INTERESTING) DS GRAPHS?

Showing that a graph is DS is usually hard. It requires extracting so much information from ϕ(G) that no other graph is consistent with all of it. Sometimes it is easy — the only graph with n

3

• triangles is Kn — but

then it is not very interesting. In general, there are results only for trivial cases, almost-trivial cases,

• r for special highly-structured graphs (such as distance-regular

graphs).

PROPORTION OF DS GRAPHS

So what is the proportion of DS graphs? Does the relative ease of constructing cospectral graphs mean that they dominate? Or is this just a reflection of the inherent asymmetry in showing that a graph is not DS compared to showing that it is DS?

COMPUTATIONAL EVIDENCE

n DS Graphs

• No. Graphs

% DS 5 32 34 94.12 6 146 156 93.59 7 934 1044 89.46 8 10624 12346 86.05 9 223629 274668 81.42 10 9444562 12005168 78.67 11 803666188 1018997864 78.87 Percentage of DS graphs on 5–11 vertices

GRAPHING THE EVIDENCE

0.5 0.6 0.7 0.8 0.9 1 5 6 7 8 9 10 11 12

Proportion of DS graphs on 5 − 12 vertices

GRAPHING THE EVIDENCE

0.5 0.6 0.7 0.8 0.9 1 5 6 7 8 9 10 11 12

Proportion of DS graphs on 5 − 12 vertices

PICK A HORSE!

Which is true?

I Almost all graphs are DS I Almost all graphs are not DS I None of the above

TREE DATA

n Trees DS trees n Trees DS trees 5 3 3 6 6 6 7 11 11 8 23 21 9 47 37 10 106 98 11 235 175 12 551 432 13 1301 886 14 3159 2333 15 7741 5271 16 19320 14074 17 48629 33685 18 123867 91520 19 317955 233837 20 823065 642676 21 2144505 1686337 22 5623756 4625719 23 14828074 12341072 24 39299897 33773440

Trees: http://oeis.org/A000055 DS trees: not in OEIS

GRAPHING DS TREES

5 10 15 20 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Proportion of DS trees on 5–24 vertices

VARIATIONS ON A THEME

The adjacency matrix is not the only matrix associated with a graph,

I If D = diag(d0, d1, . . . , dn−1) then the Laplacian matrix is

L = D A

I Pick your favourite α, β, γ and define a generalized adjacency

matrix by b A = αI + βA + γ(J I A).

I Numerous other variants

DIFFERENT POLYNOMIALS — SAME QUESTIONS

For each type of matrix, indeed for each type of polynomial P, the same sorts of questions arise:

I What properties of a graph are reflected in P? I When are two graphs co-P? I What graphs are determined by P? I What proportion of graphs are DP?

THANKS FOR LISTENING!