1 / n Expansion for the Number of Matchings on Regular Graphs and Monomer-Dimer Entropy

Abstract

Using a 1 / n expansion, that is an expansion in descending powers of n, for the number of matchings in regular graphs with 2n vertices, we study the monomer-dimer entropy for two classes of graphs. We study the difference between the extensive monomer-dimer entropy of a random r-regular graph G (bipartite or not) with 2n vertices and the average extensive entropy of r-regular graphs with 2n vertices, in the limit $n \rightarrow \infty$. We find a series expansion for it in the numbers of cycles; with probability 1 it converges for dimer density $p < 1$ and, for G bipartite, it diverges as $|\mathrm{ln}(1-p)|$ for $p \rightarrow 1$. In the case of regular lattices, we similarly expand the difference between the specific monomer-dimer entropy on a lattice and the one on the Bethe lattice; we write down its Taylor expansion in powers of p through the order 10, expressed in terms of the number of totally reducible walks which are not tree-like. We prove through order 6 that its expansion coefficients in powers of p are non-negative.

Appendices

Appendix A: $a_h$ Coefficients of $M_j$

Since $a_k$ has degree at most 2k in j, from Eq. (18) it follows that $q_k$ has degree at most $k-1$. Let

$$\begin{aligned} q_k = \sum _{h=0}^{k-1} q_{kh} j^h \end{aligned}$$

(42)

Examining the expressions computed recursively for $a_k$ for generic r through $k=20$, we find that the four highest coefficients of $q_k$ have the form

$$\begin{aligned} \begin{array}{ll} q_{k,k-1} =&{} \frac{\alpha ^k}{k!}, \qquad k \ge 1 \\ q_{k,k-2} =&{} q_{2,0} \frac{\alpha ^{k-2}}{(k-2)!}, \qquad k \ge 2 \\ q_{k,k-3} =&{} \bigg (-\frac{\alpha }{6}(1-\frac{1}{k-3}) (\frac{2}{r}-1)(\frac{1}{r}-1)^2 + q_{4,1} \bigg )\frac{\alpha ^{k-4}}{(k-4)!}, \qquad k \ge 4 \\ q_{k,k-4} =&{} \bigg ((k-6)\frac{\alpha }{6}q_{2,0}(\frac{2}{r}-1)(\frac{1}{r}-1)^2 + 2q_{6,2} + \frac{1}{6}(k^2-9k+18)q_{2,0}^3\bigg ) \\ &{} \frac{\alpha ^{k-6}}{(k-4)(k-5)(k-6)!}, \qquad k \ge 6 \end{array} \end{aligned}$$

(43)

where $\alpha \equiv \frac{1}{2r} - 1$ and

$$\begin{aligned} \begin{array}{ll} q_{2,0} =&{} \frac{7}{24 r^2} - \frac{1}{2r} + \frac{1}{6} \\ q_{4,1} =&{} \frac{241}{1152 r^4} - \frac{43}{48 r^3} + \frac{193}{144 r^2} - \frac{5}{6 r} + \frac{13}{72} \\ q_{6,2} =&{} \frac{48677}{414720 r^6} - \frac{9163}{11520 r^5} + \frac{74579}{34560 r^4} - \frac{863}{288 r^3} + \frac{19397}{8640 r^2} - \frac{619}{720 r} + \frac{851}{6480} \end{array} \end{aligned}$$

(44)

Equation (43) has been tested till $k=100$ for $r=1,\ldots , 10$ using $a_k$ computed with Eq. (19).

Here we write down the first five $q_k$.

$$\begin{aligned} \begin{array}{ll} q_1 &{}= -1 + \frac{1}{2 \, r} \\ q_2 &{}= \frac{1}{2} \, j + \frac{1}{6} -\frac{j + 1}{2 \, r} + \frac{3 \, j + 7}{24 \, r^{2}} \\ q_3 &{}= -\frac{1}{6} \, j^{2} - \frac{1}{6} \, j - \frac{1}{6} + \frac{3 \, j^{2} + 7 \, j + 8}{12 \, r} - \frac{3 \, j^{2} + 13 \, j + 20}{24 \, r^{2}} + \frac{j^{2} + 7 \, j + 16}{48 \, r^{3}} \\ q_4 &{}= \frac{1}{24} \, j^{3} + \frac{1}{12} \, j^{2} + \frac{13}{72} \, j + \frac{37}{180} -\frac{j^{3} + 4 \, j^{2} + 10 \, j + 13}{12 \, r} + \frac{9 \, j^{3} + 60 \, j^{2} + 193 \, j + 298}{144 \, r^{2}} \\ &{}\quad -\frac{j^{3} + 10 \, j^{2} + 43 \, j + 82}{48 \, r^{3}} + \frac{15 \, j^{3} + 210 \, j^{2} + 1205 \, j + 2978}{5760 \, r^{4}} \\ q_5 &{}= -\frac{1}{120} \, j^{4} - \frac{1}{36} \, j^{3} - \frac{7}{72} \, j^{2} - \frac{7}{30} \, j - \frac{3}{10} \\ &{}\quad + \frac{15 \, j^{4} + 90 \, j^{3} + 365 \, j^{2} + 994 \, j + 1416}{720 \, r} - \frac{3 \, j^{4} + 28 \, j^{3} + 142 \, j^{2} + 451 \, j + 726}{144 \, r^{2}} \\ &{}\quad +\frac{3 \, j^{4} + 40 \, j^{3} + 259 \, j^{2} + 982 \, j + 1824}{288 \, r^{3}} - \frac{15 \, j^{4} + 270 \, j^{3} + 2225 \, j^{2} + 10258 \, j + 22512}{5760 \, r^{4}} \\ &{}\quad +\frac{3 \, j^{4} + 70 \, j^{3} + 725 \, j^{2} + 4098 \, j + 10944}{11520 \, r^{5}} \end{array} \end{aligned}$$

(45)

Appendix B: Coefficients $d_h$ for the Entropy for Regular Lattice

Let $z \equiv \frac{1}{r}$. For notational simplicity, here $\epsilon _s$ is the constant $\hat{\epsilon }_s$ in the text.

The coefficient $d_h$ in Eq. (40) are given, through order 10, by

$$\begin{aligned} d_3= & {} -\frac{1}{12} \, \epsilon _{3} z^{3} \nonumber \\ d_4= & {} \frac{1}{16} \, {\left( 4 \, \epsilon _{3} + \epsilon _{4}\right) } z^{4} - \frac{1}{2} \, \epsilon _{3} z^{3} \nonumber \\ d_5= & {} -\frac{1}{16} \, \epsilon _{3}^{2} z^{6} - \frac{1}{20} \, {\left( 5 \, \epsilon _{3} + 5 \, \epsilon _{4} + \epsilon _{5}\right) } z^{5} + \frac{1}{2} \, {\left( 3 \, \epsilon _{3} + \epsilon _{4}\right) } z^{4} - \frac{7}{4} \, \epsilon _{3} z^{3} \nonumber \\ d_6= & {} \frac{1}{16} \, {\left( 7 \, \epsilon _{3}^{2} + 2 \, \epsilon _{3} \epsilon _{4}\right) } z^{7} - \frac{1}{24} \, {\left( 21 \, \epsilon _{3}^{2} - 2 \, \epsilon _{3} - 9 \, \epsilon _{4} - 6 \, \epsilon _{5} - \epsilon _{6}\right) } z^{6} - \frac{1}{2} \, {\left( 3 \, \epsilon _{3} +4 \, \epsilon _{4} + \epsilon _{5}\right) } z^{5}\nonumber \\&+\, \frac{3}{4} \, {\left( 7 \, \epsilon _{3} + 3 \, \epsilon _{4}\right) } z^{4} - \frac{14}{3} \, \epsilon _{3} z^{3}\nonumber \\ d_7= & {} -\frac{1}{12} \, \epsilon _{3}^{3} z^{9} - \frac{1}{16} \, {\left( 20 \, \epsilon _{3}^{2} + 16 \, \epsilon _{3} \epsilon _{4} + \epsilon _{4}^{2} + 2 \, \epsilon _{3} \epsilon _{5}\right) } z^{8} \nonumber \\&+\, \frac{1}{28} \, {\left( 168 \, \epsilon _{3}^{2} + 56 \, \epsilon _{3} \epsilon _{4} - 7 \, \epsilon _{4} - 14 \, \epsilon _{5} - 7 \, \epsilon _{6} - \epsilon _{7}\right) } z^{7} \nonumber \\&-\, \frac{1}{2} \, {\left( 13 \, \epsilon _{3}^{2} - \epsilon _{3} - 6 \, \epsilon _{4} - 5 \, \epsilon _{5} - \epsilon _{6}\right) } z^{6} - \frac{1}{4} \, {\left( 21 \, \epsilon _{3} + 36 \, \epsilon _{4} + 11 \, \epsilon _{5}\right) } z^{5} + \frac{1}{2} \, {\left( 28 \, \epsilon _{3} + 15 \, \epsilon _{4}\right) } z^{4} - \frac{21}{2} \, \epsilon _{3} z^{3} \nonumber \\ d_8= & {} \frac{3}{32} \, {\left( 10 \, \epsilon _{3}^{3} + 3 \, \epsilon _{3}^{2} \epsilon _{4}\right) } z^{10} - \frac{1}{16} \, {\left( 30 \, \epsilon _{3}^{3} - 30 \, \epsilon _{3}^{2} - 54 \, \epsilon _{3} \epsilon _{4} - 9 \, \epsilon _{4}^{2} - 18 \, \epsilon _{3} \epsilon _{5} - 2 \, \epsilon _{4} \epsilon _{5} - 2 \, \epsilon _{3} \epsilon _{6}\right) } z^{9} \nonumber \\&-\, \frac{1}{32} \, {\left( 540 \, \epsilon _{3}^{2} + 504 \, \epsilon _{3} \epsilon _{4} + 36 \, \epsilon _{4}^{2} + 72 \, \epsilon _{3} \epsilon _{5} - 2 \, \epsilon _{4} - 16 \, \epsilon _{5} - 20 \, \epsilon _{6} - 8 \, \epsilon _{7} - \epsilon _{8}\right) } z^{8} \nonumber \\&+\, \frac{1}{8} \, {\left( 351 \, \epsilon _{3}^{2} + 135 \, \epsilon _{3} \epsilon _{4} - 16 \, \epsilon _{4} - 40 \, \epsilon _{5} - 24 \, \epsilon _{6} - 4 \, \epsilon _{7}\right) } z^{7}\nonumber \\&- \frac{1}{8} \, {\left( 273 \, \epsilon _{3}^{2} - 14 \, \epsilon _{3} - 108 \, \epsilon _{4} - 110 \, \epsilon _{5} - 26 \, \epsilon _{6}\right) } z^{6} \nonumber \\&-\, {\left( 14 \, \epsilon _{3} + 30 \, \epsilon _{4} + 11 \, \epsilon _{5}\right) } z^{5} + \frac{3}{8} \, {\left( 84 \, \epsilon _{3} + 55 \, \epsilon _{4}\right) } z^{4} - 21 \, \epsilon _{3} z^{3} \nonumber \\ d_9= & {} -\frac{55}{384} \, \epsilon _{3}^{4} z^{12} - \frac{5}{48} \, {\left( 44 \, \epsilon _{3}^{3} + 33 \, \epsilon _{3}^{2} \epsilon _{4} + 3 \, \epsilon _{3} \epsilon _{4}^{2} + 3 \, \epsilon _{3}^{2} \epsilon _{5}\right) } z^{11} \nonumber \\&+\, \frac{1}{16} \, \Bigg (330 \, \epsilon _{3}^{3} + 110 \, \epsilon _{3}^{2} \epsilon _{4} - 25 \, \epsilon _{3}^{2} - 100 \, \epsilon _{3} \epsilon _{4} - 35 \, \epsilon _{4}^{2} - 70 \, \epsilon _{3} \epsilon _{5} - 20 \, \epsilon _{4} \epsilon _{5} \nonumber \\&-\, \epsilon _{5}^{2} - 20 \, \epsilon _{3} \epsilon _{6} - 2 \, \epsilon _{4} \epsilon _{6} - 2 \, \epsilon _{3} \epsilon _{7}\Bigg ) z^{10} \nonumber \\&-\, \frac{1}{144} \, \Bigg (3135 \, \epsilon _{3}^{3} - 3600 \, \epsilon _{3}^{2} - 7560 \, \epsilon _{3} \epsilon _{4} - 1440 \, \epsilon _{4}^{2} - 2880 \, \epsilon _{3} \epsilon _{5} - 360 \, \epsilon _{4} \epsilon _{5} - 360 \, \epsilon _{3} \epsilon _{6}\nonumber \\&+ 36 \, \epsilon _{5} + 120 \, \epsilon _{6} + 108 \, \epsilon _{7} +36 \, \epsilon _{8} + 4 \, \epsilon _{9}\Bigg ) z^{9} \nonumber \\&-\, \frac{1}{8} \, {\left( 975 \, \epsilon _{3}^{2} + 1050 \, \epsilon _{3} \epsilon _{4} + 85 \, \epsilon _{4}^{2} + 170 \, \epsilon _{3} \epsilon _{5} - 4 \, \epsilon _{4} - 40 \, \epsilon _{5} - 60 \, \epsilon _{6} - 28 \, \epsilon _{7} - 4 \, \epsilon _{8}\right) } z^{8} \nonumber \\&+\, \frac{1}{4} \, {\left( 910 \, \epsilon _{3}^{2} + 400 \, \epsilon _{3} \epsilon _{4} - 36 \, \epsilon _{4} - 110 \, \epsilon _{5} - 78 \, \epsilon _{6} - 15 \, \epsilon _{7}\right) } z^{7} \nonumber \\&-\, \frac{1}{48} \, {\left( 6825 \, \epsilon _{3}^{2} - 224 \, \epsilon _{3} - 2160 \, \epsilon _{4} - 2640 \, \epsilon _{5} - 728 \, \epsilon _{6}\right) } z^{6} \nonumber \\&-\, \frac{1}{4} \, {\left( 126 \, \epsilon _{3} + 330 \, \epsilon _{4} + 143 \, \epsilon _{5}\right) } z^{5} + \frac{9}{2} \, {\left( 14 \, \epsilon _{3} + 11 \, \epsilon _{4}\right) } z^{4} - \frac{77}{2} \, \epsilon _{3} z^{3} \end{aligned}$$

(46)

$$\begin{aligned} d_{10}= & {} \frac{11}{64} \, {\left( 13 \, \epsilon _{3}^{4} + 4 \, \epsilon _{3}^{3} \epsilon _{4}\right) } z^{13} - \frac{11}{96} \, \Big (39 \, \epsilon _{3}^{4} - 112 \, \epsilon _{3}^{3} - 162 \, \epsilon _{3}^{2} \epsilon _{4} - 36 \, \epsilon _{3} \epsilon _{4}^{2} - \epsilon _{4}^{3}\nonumber \\&- 36 \, \epsilon _{3}^{2} \epsilon _{5} - 6 \, \epsilon _{3} \epsilon _{4} \epsilon _{5} - 3 \, \epsilon _{3}^{2} \epsilon _{6}\Big ) z^{12} \nonumber \\&-\, \frac{1}{16} \, (1584 \, \epsilon _{3}^{3} + 1320 \, \epsilon _{3}^{2} \epsilon _{4} + 132 \, \epsilon _{3} \epsilon _{4}^{2} + 132 \, \epsilon _{3}^{2} \epsilon _{5} - 11 \, \epsilon _{3}^{2} - 110 \, \epsilon _{3} \epsilon _{4} - 77 \, \epsilon _{4}^{2} \nonumber \\&-\, 154 \, \epsilon _{3} \epsilon _{5} - 88 \, \epsilon _{4} \epsilon _{5} - 11 \, \epsilon _{5}^{2}-88 \, \epsilon _{3} \epsilon _{6} - 22 \, \epsilon _{4} \epsilon _{6} - 2 \, \epsilon _{5} \epsilon _{6} - 22 \, \epsilon _{3} \epsilon _{7} - 2 \, \epsilon _{4} \epsilon _{7} - 2 \, \epsilon _{3} \epsilon _{8}) z^{11} \nonumber \\&+\, \frac{1}{40} \, (9405 \, \epsilon _{3}^{3} + 3465 \, \epsilon _{3}^{2} \epsilon _{4} - 825 \, \epsilon _{3}^{2} - 3850 \, \epsilon _{3} \epsilon _{4} - 1540 \, \epsilon _{4}^{2} - 3080 \, \epsilon _{3} \epsilon _{5} - 990 \, \epsilon _{4} \epsilon _{5} \nonumber \\&-\, 55 \, \epsilon _{5}^{2} - 990 \, \epsilon _{3} \epsilon _{6}-110 \, \epsilon _{4} \epsilon _{6} - 110 \, \epsilon _{3} \epsilon _{7} + \epsilon _{10} + 2 \, \epsilon _{5} + 25 \, \epsilon _{6} + 50 \, \epsilon _{7} + 35 \, \epsilon _{8} + 10 \, \epsilon _{9}) z^{10} \nonumber \\&-\, \frac{1}{24} \, (4180 \, \epsilon _{3}^{3} - 4290 \, \epsilon _{3}^{2} - 10395 \, \epsilon _{3} \epsilon _{4} - 2244 \, \epsilon _{4}^{2} - 4488 \, \epsilon _{3} \epsilon _{5} - 627 \, \epsilon _{4} \epsilon _{5} - 627 \, \epsilon _{3} \epsilon _{6} \nonumber \\&+\, 60 \, \epsilon _{5} + 240 \, \epsilon _{6}+ 252 \, \epsilon _{7} + 96 \, \epsilon _{8} + 12 \, \epsilon _{9}) z^{9} \nonumber \\&-\, \frac{1}{8} \, {\left( 5005 \, \epsilon _{3}^{2} + 6160 \, \epsilon _{3} \epsilon _{4} + 561 \, \epsilon _{4}^{2} + 1122 \, \epsilon _{3} \epsilon _{5} - 18 \, \epsilon _{4} - 220 \, \epsilon _{5} - 390 \, \epsilon _{6} - 210 \, \epsilon _{7} - 34 \, \epsilon _{8}\right) } z^{8} \nonumber \\&+\, \frac{1}{16} \, {\left( 15015 \, \epsilon _{3}^{2} + 7480 \, \epsilon _{3} \epsilon _{4} - 480 \, \epsilon _{4} - 1760 \, \epsilon _{5} - 1456 \, \epsilon _{6} - 320 \, \epsilon _{7}\right) } z^{7} \nonumber \\&-\, \frac{1}{8} \, {\left( 4004 \, \epsilon _{3}^{2} - 84 \, \epsilon _{3} - 990 \, \epsilon _{4} - 1430 \, \epsilon _{5} - 455 \, \epsilon _{6}\right) } z^{6} - \frac{1}{10} \, {\left( 630 \, \epsilon _{3} + 1980 \, \epsilon _{4} + 1001 \, \epsilon _{5}\right) } z^{5} \nonumber \\&+\, \frac{33}{4} \, {\left( 14 \, \epsilon _{3} + 13 \, \epsilon _{4}\right) } z^{4} - 66 \, \epsilon _{3} z^{3} \end{aligned}$$

(47)

Appendix C: Proof of a Conjecture in [6] in Some Cases

The $n\times n$ reduced adjacency matrix A of a bipartite r-regular graph G with 2n vertices has entries 0 and 1, and the sums of the rows and of the columns equals r.

The uniform distribution of the labelled graphs corresponds to the expectation E.

The number or i-matchings on G is $m_i(G) = {{\mathrm{perm}}}_i(A)$, defined as the sum of the permanents of the $i\times i$ minors of A.

$A_i \equiv E({{\mathrm{perm}}}_i(A))$ is the average number of i-matchings.

Another expectation $E_2$ is defined by the second measure in Section 4 of [9] for the non negative integer matrices, with

$$\begin{aligned} A_i^{E_2} \equiv E_2({{\mathrm{perm}}}_i(A)) = \left( {\begin{array}{c}n\\ i\end{array}}\right) ^2 \frac{r^{2i} i! (rn-i)!}{(rn)!} \end{aligned}$$

(48)

It is the average number of regular bipartite multigraphs under the expectation $E_2$.

In [6] the following quantity is defined

$$\begin{aligned} \mathcal{C}_i= \bigg ( \frac{1}{r^i} \frac{n^i (n-i)!}{n!} \bigg ) \cdot \sum _{k=0}^ i (-1)^k \left( {\begin{array}{c}n-i+k\\ k\end{array}}\right) ^2 k! (\frac{r}{n})^k A_{i-k} \end{aligned}$$

(49)

with $\mathcal{C}_0=1$ and $\mathcal{C}_1 =0$.

From this the following cluster expansion is constructed

$$\begin{aligned} \sum _i T_i x^i = \mathrm{ln}(\sum _i \mathcal{C}_i x^i) \end{aligned}$$

(50)

Similarly using Eq. (49), the quantities $C_i^{E_2}$ are computed in terms of $A_i^{E_2}$; from this one gets $T_i^{E_2}$ using Eq. (50).

In [6] it is conjectured that

$$\begin{aligned} Q_i(r) \equiv \lim _{n\rightarrow \infty } \frac{1}{n} T_i(n) = \lim _{n\rightarrow \infty } \frac{1}{n} T_i^{E_k}(n,r), \quad k=1,2 \end{aligned}$$

(51)

Using Eqs. (48, 49, 50) one can compute the first few $Q_i$. In [6] the conjecture has been proven for $i \le 3$; some numerical evidence has been given for $4 \le i \le 25$.

We have proven this conjecture for $i \le 7$.

For $i \le 7$, $m_i$ is linear in the $\epsilon$’s. Since $\epsilon _s$ is linear in the contributors, $E(\epsilon _s)$ is linear in the average of the contributors, so it can be computed modulo $o(n^0)$ using $E(C_s) = \frac{1}{s}(r-1)^s + o(n^0)$ and the fact that the polycyclic contributors are $o(n^0)$ [19]. For $i \le 7$, $\epsilon _i$ is given in Eq. (6).

It is easy to see from Eqs. (49, 50) that $T_i$ is linear in the average of contributors; expanding $T_i$ in 1 / n expansion, keeping the average of the contributors as independent variables, it has the form

$$\begin{aligned} T_i = n Q_i + B_i \end{aligned}$$

(52)

where $Q_i$ is the one found in [6], and $B_i$ depends on r and is linear in the average of contributors, plus $o(n^0)$ terms. For instance

$$\begin{aligned} B_4 = -\frac{r^{4} - 8 \, r^{3} + 37 \, r^{2} - 30 \, r - 4 \, E(C_{4})}{4 \, r^{4}} + o(n^0) \end{aligned}$$

(53)

It follows that $\lim _{n \rightarrow \infty }B_i$ is a function of r for $i \le 7$.