T HE C HARACTERISTIC P OLYNOMIAL OF A G RAPH Gordon Royle Centre for the Mathematics of Symmetry & Computation School of Mathematics & Statistics University of Western Australia Dagstuhl, June 2016
A USTRALIA
P ERTH
U NIVERSITY OF W ESTERN A USTRALIA
O UTLINE I NTRODUCTION S OME B ASICS R ECONSTRUCTION DS GRAPHS
G RAPHS The Petersen graph is a I cubic graph, I with 10 vertices , I and 15 edges .
C HARACTERISTIC 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).
E XAMPLE 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 A ( P ) = 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 0 1 0 ϕ ( P ; x ) = det ( xI � A ( P )) = ( x � 3 )( x � 1 ) 5 ( x + 2 ) 4
T HE 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 , 1 5 , � 2 4 } .
S PECTRAL GRAPH THEORY Spectral graph theory is the study of the relationship between graphical properties and graph spectra.
S OME 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
O UTLINE I NTRODUCTION S OME B ASICS R ECONSTRUCTION DS GRAPHS
W HAT DOES IT COUNT ? If G is a graph on n vertices then the coefficient of x n − r in ϕ ( G ; x ) is X ( � 1 ) comp ( H ) 2 cyc ( H ) 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 .
W HY ? We know that for any n ⇥ n matrix N , X Y det ( N ) = sgn ( σ ) N v σ ( v ) v σ ∈ Sym ( n ) In order for σ to contribute, the product must be non-zero so N v σ ( v ) 6 = 0 for all v 2 { 1 , . . . , n } . Applying this to xI � A shows that { v σ ( 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 ) − 1 x 0 − 1 0 − 1 0 0 0 0 − 1 0 x 0 − 1 0 − 1 0 0 0 − 1 − 1 0 x 0 0 0 − 1 0 0 − 1 − 1 0 x 0 − 1 0 0 − 1 0 − 1 − 1 − 1 0 x 0 0 0 0 − 1 − 1 − 1 0 0 0 0 x 0 0 − 1 − 1 0 − 1 0 0 0 x − 1 0 0 x 0 0 − 1 0 0 0 − 1 − 1 0 − 1 0 0 0 − 1 0 0 0 − 1 x − 1 0 0 0 0 − 1 − 1 0 0 x
I MMEDIATE 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 x n − even so ϕ ( G ; x ) = p ( x 2 ) or x p ( x 2 ) for some polynomial p , so the spectrum is symmetric about 0.
F UNDAMENTAL 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 ?
C OSPECTRALITY 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 .
T HE S ALTIRE P AIR ϕ ( x ) = x 3 ( x 2 � 4 )
T HE S ALTIRE P AIR ϕ ( x ) = x 3 ( x 2 � 4 ) The Saltire
W HAT 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
W HAT 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
W ALKS These results follow from a few simple facts: I The vw -entry of A 2 counts the 2-step walks from v to w . I The vw -entry of A k counts the k -step walks from v to w . I The total number of closed walks of length k is tr A k . I The trace of A k is P λ λ k .
A ND REGULARITY The regularity of a graph can be determined from its spectrum: A graph is k -regular if and only if λ 1 = k and X λ 2 = kn . λ This follows because for a non-regular graph k min < ¯ k < λ 1 < k max .
O UTLINE I NTRODUCTION S OME B ASICS R ECONSTRUCTION DS GRAPHS
T HE 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?
T HE RECONSTRUCTION CONJECTURE Ulam’s reconstruction conjecture is that non-isomorphic graphs 1 have different decks: deck ( G ) = deck ( H ) ) G ' H . A graphical parameter is reconstructible if its value can be determined from the deck. 1 on at least 3 vertices
C HARACTERISTIC POLYNOMIAL Elementary linear algebra shows us that X d dx ( ϕ ( G ; x )) = ϕ ( G \ v ; x ) v ∈ V ( G ) 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 ) .
T UTTE In his famous paper “All the King’s Horses” , Bill Tutte showed that a variety of graph parameters are reconstructible.
R ECONSTRUCTIBLE 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 , . . . )
T HE 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 ) } x 4 � 3 x 2 x 4 � 3 x 2 x 4 � 3 x 2 x 4 � 3 x 2 x 4 Can the characteristic polynomial of G be reconstructed from the polynomial deck of G ? This is an open problem .
T WO PARTIAL RESULTS T HEOREM (H AGOS 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 . T HEOREM (C VETKOVI ´ C AND L EPOVI ´ C 1998) The characteristic polynomial of a tree T is reconstructible from the polynomial deck of T .
O UTLINE I NTRODUCTION S OME B ASICS R ECONSTRUCTION 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.
S MALL G RAPHS Even with extra conditions (in this case, regularity) there are many cospectral small graphs. The smallest connected cubic cospectral graphs
S TRONGLY 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 bucketloads 2 of strongly regular graphs — for example, 11 billion 57-vertex SRGs from Steiner Triple Systems with 19 points. 2 a technical term
G ODSIL /M C K AY 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 }
G ODSIL /M C K AY SWITCHING Then form two matrices B � B � b N N A 1 = A 2 = N T b N T C C where J is an all-ones matrix and b N is obtained from N by exchanging ones and zeros in all the columns of weight b / 2. Then A 1 and A 2 are similar matrices.
P ROOF Let 2 � b J b � I b 0 Q = 0 I c Then Q 2 = I b + c and QA 1 Q = A 2
G RAPHICALLY Graphically this gives a partition with the “none, half or all” property. and cospectral graphs are obtained by switching on the b / 2-vertices.
E XAMPLE These both have ϕ ( x ) = ( x + 1 ) 2 ( x 8 − 2 x 7 − 16 x 6 + 16 x 5 + 72 x 4 − 42 x 3 − 96 x 2 + 44 x + 7 ) But are they isomorphic? (no)
S CHWENK ’ S FAMOUS RESULT In the 1970s, Allen Schwenk proved the following famous result: T HEOREM Almost all trees are cospectral. If T is the tree then it is routine to check that ϕ ( T \ ) = ϕ ( T \ )
S CHWENK ’ 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 ) � x ϕ ( S \ v ) ϕ ( T \ v ) I Almost all trees contain the 9-vertex tree from the previous slide
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