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Study of dimer–monomer on the generalized Hanoi graph

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Abstract

We study the number of dimer–monomers \(M_d(n)\) on the Hanoi graphs \(H_d(n)\) at stage n with dimension d equal to 3 and 4. The entropy per site is defined as \(z_{H_d}=\lim _{v \rightarrow \infty } \ln M_d(n)/v\), where v is the number of vertices on \(H_d(n)\). We obtain the lower and upper bounds of the entropy per site, and the convergence of these bounds approaches to zero rapidly when the calculated stage increases. The numerical values of \(z_{H_d}\) for \(d=3, 4\) are evaluated to more than a hundred digits correct. Using the results with d less than or equal to 4, we predict the general form of the lower and upper bounds for \(z_{H_d}\) with arbitrary d.

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Acknowledgements

This research of S.-C.C. was supported in part by the MOST Grant 107-2515-S-006-002.

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Correspondence to Shu-Chiuan Chang.

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Communicated by Maria Aguieiras de Freitas.

A Proof of the monotonicity of \(\omega _3(n)\) and \(\alpha _3(n)\)

A Proof of the monotonicity of \(\omega _3(n)\) and \(\alpha _3(n)\)

We shall show that \(\omega _3(n)\) is an ascending function and \(\alpha _3(n)\) a descending function here. Using abc to denote \(\alpha _3(n)-\beta _3(n)\), \(\beta _3(n)-\gamma _3(n)\), \(\gamma _3(n)-\omega _3(n)\), respectively, and the definition given in Eq. (21), we find that \(\omega _3(n+1)\) is always larger than \(\omega _3(n)\) as follows.

$$\begin{aligned}&\omega _3(n+1)-\omega _3(n) =\frac{\omega _3(n)}{E_3(n)}(D_3(n)-E_3(n))\nonumber \\&\quad =\frac{\omega _3(n)}{E_3(n)}[(64a+64c+64b)\omega ^{11}\nonumber \\&\qquad +(256ab+768bc+384b+512c^2+384c+256b^2+288a+512ac)\omega ^{10}\nonumber \\&\qquad +(2688c^2+1792c^3+3584bc^2+1152c+2176b^2c+3840bc+2016ac+624a\nonumber \\&\qquad +384b^3+384ab^2+864ab+1792ac^2+1792abc+1104b+1152b^2)\omega ^9\nonumber \\&\qquad +(2352b^2+7680b^2c^2+8064c^3+9120bc+5376abc^2+8960bc^3+256ab^3+256b^4\nonumber \\&\qquad +1968b+2304ab^2c+840a+5184abc+1152b^3+3584ac^3+6912c^2+1392ab\nonumber \\&\qquad +8064b^2c+864ab^2+2232c+6048ac^2+3584c^4+2560cb^3+3744ac+14976bc^2)\omega ^8\nonumber \\&\qquad +(1392b^3+2808b^2+13296b^2c+13200bc+1416ab+288ab^3+9360ac^2+6960abc\nonumber \\&\qquad +4320ab^2c+912ab^2+1280ab^3c+384b^4+6144cb^3+64ab^4+1344b^4c+4224ac\nonumber \\&\qquad +64b^5+3072c+2400b+768a+11184c^2+17280c^3+13440c^4+29040bc^2+30720bc^3\nonumber \\&\qquad +23040b^2c^2+10080ac^3+12960abc^2+4480c^5+13440bc^4+14720b^2c^3+7040b^3c^2\nonumber \\&\qquad +4480ac^4+8960abc^3+5760ab^2c^2)\omega ^7\nonumber \\&\qquad +(864b^3+2148b^2+12432b^2c+256ab^4c+12720bc+1020ab+144ab^3+8520ac^2\nonumber \\&\qquad +5736abc+3648ab^2c+600ab^2+1152ab^3c+144b^4+5760cb^3+1536b^4c+256b^5c\nonumber \\&\qquad +3168ac+3132c+2088b+492a+12432c^2+22440c^3+23040c^4+33216bc^2\nonumber \\&\qquad +47040bc^3+29664b^2c^2+12480ac^3+13920abc^2+3584c^6+13440c^5+12544bc^5\nonumber \\&\qquad +16640b^2c^4+10240b^3c^3+36480bc^4+34560b^2c^3+13056b^3c^2+2816b^4c^2+10080ac^4\nonumber \\&\qquad +17280abc^3+8640ab^2c^2+8960abc^4+7680ab^2c^3+2560ab^3c^2+3584ac^5)\omega ^6\nonumber \\&\qquad +(384ab^4c^2+300b^3+1086b^2+7188b^2c+8436bc+552ab+24ab^3+4992ac^2\nonumber \\&\qquad +3252abc+1872ab^2c+252ab^2+432ab^3c+24b^4+2688cb^3+432b^4c+384b^5c^2\nonumber \\&\qquad +1638ac+2404c+1312b+220a+9792c^2+19008c^3+22560c^4+1792c^7\nonumber \\&\qquad +24192bc^2+7168bc^6+11136b^2c^5+8320b^3c^4+40224bc^3+20592b^2c^2+8640ac^3\nonumber \\&\qquad +8784abc^2+8064c^6+17280c^5+25344bc^5+28800b^2c^4+5376abc^5+5760ab^2c^4\nonumber \\&\qquad +2560ab^3c^3+13824b^3c^3+1792ac^6+2944b^4c^3+41520bc^4+32736b^2c^3+8928b^3c^2\nonumber \\&\qquad +2304b^4c^2+9360ac^4+13920abc^3+5472ab^2c^2+12960abc^4+8640ab^2c^3\nonumber \\&\qquad +1728ab^3c^2+6048ac^5)\omega ^5 \end{aligned}$$
$$\begin{aligned}&\qquad +(256b^5c^3+1792abc^6+2304ab^2c^5+1280ab^3c^4+512ac^7+256ab^4c^3+60b^3\nonumber \\&\qquad +364b^2+2688b^2c+3870bc+226ab+1962ac^2+1326abc+600ab^2c+60ab^2\nonumber \\&\qquad +72ab^3c+744cb^3+72b^4c+590ac+1386c+588b+66a+5462c^2+10584c^3\nonumber \\&\qquad +13152c^4+2688c^7+11460bc^2+9600bc^6+12672b^2c^5+7296b^3c^4+20256bc^3\nonumber \\&\qquad +8364b^2c^2+512c^8+1536b^4c^4+2304bc^7+4096b^2c^6+3584b^3c^5+3648ac^3\nonumber \\&\qquad +3636abc^2+6912c^6+11400c^5+19104bc^5+17904b^2c^4+5184abc^5+4320ab^2c^4\nonumber \\&\qquad +1152ab^3c^3+6144b^3c^3+2016ac^6+1536b^4c^3+23856bc^4+15264b^2c^3+2880b^3c^2\nonumber \\&\qquad +432b^4c^2+4440ac^4+6096abc^3+2016ab^2c^2+6960abc^4+3648ab^2c^3+432ab^3c^2\nonumber \\&\qquad +3744ac^5)\omega ^4\nonumber \\&\qquad +(864abc^6+864ab^2c^5+288ab^3c^4+288ac^7+6b^3+81b^2+670b^2c+1212bc\nonumber \\&\qquad +66ab+526ac^2+388abc+120ab^2c+6ab^2+120cb^3+147ac+588c+180b\nonumber \\&\qquad +12a+2094c^2+3718c^3+4320c^4+1152c^7+3518bc^2+3600bc^6+3888b^2c^5\nonumber \\&\qquad +1584b^3c^4+64ac^8+640b^2c^7+640b^3c^6+320b^4c^5+64b^5c^4+256abc^7+64c^9\nonumber \\&\qquad +320bc^8+5964bc^3+2118b^2c^2+384ab^2c^6+256ab^3c^5+64ab^4c^4+384c^8+384b^4c^4\nonumber \\&\qquad +1536bc^7+2304b^2c^6+1536b^3c^5+978ac^3+996abc^2+2352c^6+3648c^5+5712bc^5\nonumber \\&\qquad +4440b^2c^4+1392abc^5+912ab^2c^4+144ab^3c^3+1152b^3c^3+624ac^6+144b^4c^3\nonumber \\&\qquad +6816bc^4+3756b^2c^3+588b^3c^2+72b^4c^2+1152ac^4+1596abc^3+444ab^2c^2\nonumber \\&\qquad +1704abc^4+816ab^2c^3+72ab^3c^2+960ac^5)\omega ^3\nonumber \\&\qquad +(12b^2+111b^2c+255bc+12ab+96ac^2+81abc+12ab^2c+12cb^3+24ac\nonumber \\&\qquad +175c+34b+a+513c^2+738c^3+666c^4+24c^7+684bc^2+96bc^6+144b^2c^5\nonumber \\&\qquad +96b^3c^4+978bc^3+324b^2c^2+162ac^3+174abc^2+144c^6+396c^5+432bc^5\nonumber \\&\qquad +432b^2c^4+72abc^5+72ab^2c^4+24ab^3c^3+144b^3c^3+24ac^6+24b^4c^3+852bc^4\nonumber \\&\qquad +516b^2c^3+60b^3c^2+162ac^4+222abc^3+60ab^2c^2+192abc^4+96ab^2c^3+96ac^5)\omega ^2\nonumber \\&\qquad +(18b^2c^3+30b^2c^2+15ac^3+12ac^2+18bc^4+12abc+68c^2+33c+36bc+6c^5\nonumber \\&\qquad +b^2+30c^4+3b+75bc^2+12abc^3+2ac+ab+60bc^3+6ab^2c^2+15abc^2+6b^3c^2\nonumber \\&\qquad +63c^3+6ac^4+12b^2c)\omega \nonumber \\&\qquad +2bc^2+ac^2+b^2c+3c+3bc+abc+3c^2+c^3]>0 , \end{aligned}$$
(42)

where the inequality holds since all terms are positive. The relation \(\alpha _3(n)-\alpha _3(n+1) > 0\) can be proved similarly, such that \(\alpha _3(n)\) decreases monotonically as n increases.

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Li, W., Chang, S. Study of dimer–monomer on the generalized Hanoi graph. Comp. Appl. Math. 39, 77 (2020). https://doi.org/10.1007/s40314-020-1088-x

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Keywords

  • Dimer–monomer model
  • Hanoi graph
  • Recursion relations
  • Entropy per site

Mathematics Subject Classification

  • 05A20
  • 05C70
  • 82B20