(N, N×N) sinceGhas a DCDC, and|c(G)| = N(N−1). . .(N− |V(G)|+ 1).
Proposition 11.4.6. A term(q, P)contributes to< gr3,...,rk(M)>if and only if there is a simple graph G with a DCDC consisting of ri cycles of length i (i= 3, . . . , k) such that(q, P)∈c(G).
Proof. If (q, P)∈c(G), then any DCDC provides a partition ofqinto its cycles and hence (q, P) contributes to < gr3,...,rk(M)>. On the other hand if (q, P) contributes to < gr3,...,rk(M)>, then lettingGbe the graph with the vertices from{1, . . . , N}and the edges given byP we get thatGis simple sinceqconsists of edge-disjoint directed cycles, it has a DCDC consisting ofri cycles of length i(i= 3, . . . , k), and (q, P)∈c(G).
Proposition 11.4.7. Ifc(G)∩c(G0)6=∅, thenGis isomorphic toG0. Moreover, ifGis isomorphic to G0, thenc(G) =c(G0).
Proof. If (q, P) ∈ c(G)∩c(G0), then the construction of q induces a function between the sets of vertices of GandG0, andP gives the edges of bothG, G0. Hence they are isomorphic. The second part is true since the definition ofc(G) does not depend on ’names’ of the vertices.
As a consequence we have Theorem 11.4.8.
< gr3,...,rk(M)>= X
[G]
N(N−1). . .(N− |V(G)|+ 1)
Ne(G) ,
where the sum is over all isomorphism classes of simple graphs with at most N vertices that have a DCDC consisting ofri cycles of length i(i= 3, . . . , k).
Moreover
< gr03,...,rk(M)>= X
[G]0
N(N−1). . .(N− |V(G)|+ 1)
Ne(G) ,
where the sum is over all isomorphism classes of simple graphs with at most N vertices and with a specified DCDC consisting of ri cycles of length i (i = 3, . . . , k).
Analogous statements hold for gr, gr0, g, g0.
11.5 Calculations
The integral < g0(M) >counts all the directed cycle double covers of graphs on at mostN vertices and hence its calculation is an attractive task which need not be hopeless. We show next a curious formula forg0(M) which identifies it with an Ihara-Selberg-type function (see Theorem 11.5.3). Let us recall that
g0(M) = X
c∈A0
Y
e∈c
Me, (11.5.1)
is the generating function (with variablesMe’s) of the collections of edge-disjoint directed cycles of length at least three, in the directed graph D = D(M) = (N, N×N).
148 CHAPTER 11. GAUSSIAN MATRIX INTEGRAL METHOD
Construction of digraph D’. We first construct a directed graph D0 with the weights on the transitions between the edges. First we split each vertex of D, i.e., we replace each vertexv by new edgee(v) and we let all the edges ofD enteringv enter the initial vertex ofe(v), and all the edges ofD leavingv leave the terminal vertex ofe(v). If edgeg entersv in D then we define the weight of the transitionw(g, e(v)) =Mg. We let all the remaining transition be equal to one (see Figure 11.4, the first two parts).
Finally, for each pair g1, g2 of oppositely directed edges of D, say g1 = (uv), g2= (vu) we introduce new vertexvg and we let bothg1, g2 pass through it; equivalently, we subdivide bothg1, g2by one vertex and identify this pair of vertices into unique vertex calledvg (and thus we have new edges (uvg),(vgv) from g1 = (uv), and new edges (vvg),(vgu) from g2 = (vu)) (see Figure 11.4, the last two parts).
We let the weights of the transitions at vertex vg between g1 and g2 (i.e., between (uvg) and (vgu) and between (vvg) and (vgv)) be equal to zero, the transitions along g1 and g2 (i.e., between (uvg) and (vgv) and between (vvg) and (vgu)) be equal to one, and the transitions between (vgv) and e(v) be equal toMg1 and between (vgu) and e(u) be equal toMg2. See an example in Figure 11.5.
Figure 11.4: Construction ofe(v) andvg
Figure 11.5: An example of the construction of digraph D’
In what follows, the directed closed walk is considered not pointed. We let the weight of the directed closed walk be the product of the weights of its transitions.
Observation 11.5.1. There is a weight preserving bijection between the set of the directed cycles ofD of length at least three and the set of the closed directed
11.5. CALCULATIONS 149 walks ofD0of a non-zero weight which go through each directed edge and through each vertex vg at most once.
Proof. This follows directly from the construction ofD0.
Definition 11.5.2. We define the rotation number for each closed walk w of D0 with a non-zero weight by induction as follows: first order the directed edges of D0, say as a1, . . . , am, so that the edges e(v), v ∈ V(D) form the terminal segment. Then
(1) Ifwis a directed cycle, then we let r(w) =−1.
(2) Let w go at least twice through a directed edge. Let a be the first such edge in the fixed ordering. Hence w is a concatenation of two shorter closed walksw1, w2, both containinga. Ifa6=e(v)for somev then we let r(w) =r(w1)r(w2). If a=e(v), then we letr(w) = 0.
(3) If none of 1.,2. applies,w must go through a vertexvg (introduced in the definition of D0) at least twice. Then we again letr(w) = 0.
Theorem 11.5.3. Let g0(M)be defined as (11.5.1). Then g0(M) =Y
p
(1−r(p)w(p)),
where the product is over all aperiodic closed directed walks pin D0 and w(p) denotes the weight ofp.
To prove Theorem 11.5.3 we will need a curious lemma on coin arrangements stated below. It has been introduced by Sherman [130] in the study of 2-dimensional Ising problem.
Lemma 11.5.4 (A lemma on coin arrangements.). Suppose we have a fixed collection of N objects of which m1 are of one kind, m2 are of second kind,
· · ·, and mn are of n-th kind. Let bk be the number of exhaustive unordered arrangements of these symbols intokdisjoint, nonempty, circularly ordered sets such that no two circular orders are the same and none are periodic. For example let us have 10 coins of which 3 are pennies, 4 are nickles and 3 are quarters.
Then{(p, n),(n, p),(p, n, n, q, q, q)}is not a correct arrangement since(p, n)and (n, p)represent the same circular order. IfN >1 thenPN
i=1(−1)i+1bi= 0.
Proof of Lemma 11.5.4. The lemma follows immediately if we expand the LHS of the followingWitt Identityand collect terms where the sums of the exponents of thezi’s are the same.
Witt Identity (see [75]): Letz1, ..., zk be commuting variables. Then Y
m1,...,mk≥0
(1−z1m1...zkmk)M(m1,...,mk)= 1−z1−z2−...−zk,
where M(m1, ...., mk) is the number of different nonperiodic sequences of zi’s taken with respect to circular order.
150 CHAPTER 11. GAUSSIAN MATRIX INTEGRAL METHOD
Proof of Theorem 11.5.3. We first show that the coefficients corresponding to the products of variables where at least one Me, e 6= e(v), appears with the exponent greater than one, are all equal to zero.
Let us denoteW(p) =−r(p)w(p). LetA1be the set of all non-periodic closed walkspsuch thata1 appears inp. Eachp∈A1 has a unique factorization into words (W1, ..., Wk) each of which starts witha1and has no other appearance of a1.
LetSbe a monomial summand in the expansion ofQ
p∈A1(1+W(p)). Hence S is a product of finitely manyW(p), p∈A1.
Eachp∈A1has a unique factorization into words defined above. Each word may appear several times in the factorization ofpand also in the factorization of different non-periodic closed walks. Let B(D0) be the set-system of all the words (with repetition) appearing in the factorizations of the aperiodic closed walks ofD0.
It directly follows from Lemma 11.5.4, the lemma on coin arrangements, that the sum of all monomial summands S in the expansion of Q
p∈A1(1 +W(p)), which have the same ’coins’B(D0) of more than one element is zero. Hence the monomial summandsS which survive in the expansion ofQ
p∈A1(1 +W(p) all have B(D0) consisting of exactly one word. Hence they cannot have a1 with exponent bigger than one. Now we can repeat the same consideration for the other edges different frome(v), v∈V.
Hence the only terms of the expansion of the infinite product that survive have allMe, e6=e(v), with the exponent at most one.
We know from Observation 11.5.1 that the collections of the edge-disjoint directed cycles of length at least three inD0correspond to the collections of the directed closed aperiodic walks of D0 where each edge e 6=e(v) of D0 appears at most once; by above, these exactlyhave chance to survive.
Each term ofg0(M) may be expressed several times as a product of aperiodic closed walks ofD0,but only one such expression survives in the infinite product since if a closed walk goes through an edge e(v) or through a vertex vg more than once, its rotation is defined to be zero. Henceg0(M) is counted correctly in the infinite product.
Remark. Let us write r(p) =qrot(p), where q=−1. Without the zero values of r(p), function rot(p) is additive when we ’smoothen’ pinto directed cycles.
The integer lattice generated by the directed cycles has a basis which may be constructed e.g., from the ear-decomposition [68]; the function rot(p) may be split into contributions of the edge-transitions for the basis, and since it is a basis, it may be split also for all the directed cycles. Hence if the additivity property holds, rot(p) may be split into the contributions rot(t) of the edge-transitionst for the aperiodic closed walks. Hence
Y
p
(1−r(p)w(p)) =Y
p
(1−Y
t∈p
(−1)rot(t)w(t)).
This formula transforms the infinite product into theIhara-Selberg function. It was studied by Bass in [10] who proved that it is equal to a determinant. A combinatorial proof was given by Foata and Zeilberger in [65].
Due to the zero values of r(p) it is not clear how to split the rotation into individual edge-transitions. A determinant-type formula, perhaps non-commutative, may however exist. Moreover the Ihara-Selberg function and its
11.6. PLANAR GRAPHS WITH GIVEN DEGREE SEQUENCE 151