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Comparing Graphs of Different Sizes


We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of independent sets, among other combinatorial quantities. Our methods involve inequalities for determinants, for traces of functions of operators, and for entropy.

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[1] Aldous, D. J. and Lyons, R. (2007) Processes on unimodular random networks. Electron. J. Probab. 12 #54 14541508.
[2] Antezana, J., Massey, P. and Stojanoff, D. (2007) Jensen's inequality for spectral order and submajorization. J. Math. Anal. Appl. 331 297307.
[3] Benjamini, I., Lyons, R. and Schramm, O. (2015) Unimodular random trees. Ergodic Theory Dynam. Systems 35 359373.
[4] Bhatia, R. (1997) Matrix Analysis , Vol. 169 of Graduate Texts in Mathematics, Springer.
[5] Chung, F. R. K., Graham, R. L., Frankl, P. and Shearer, J. B. (1986) Some intersection theorems for ordered sets and graphs. J. Combin. Theory Ser. A 43 2337.
[6] Fontes, L. R. G. and Mathieu, P. (2006) On symmetric random walks with random conductances on ℤ d . Probab. Theory Rel. Fields 134 565602.
[7] Heicklen, D. and Hoffman, C. (2005) Return probabilities of a simple random walk on percolation clusters. Electron. J. Probab. 10 #8 250302.
[8] Janson, S. (2006) Conditioned Galton–Watson trees do not grow. In Fourth Colloquium on Mathematics and Computer Science: Algorithms, Trees, Combinatorics and Probabilities, DMTCS Proc. AG, pp. 331–334.
[9] Löwner, K. (1934) Über monotone Matrixfunktionen. Math. Z. 38 177216.
[10] Luczak, M. and Winkler, P. (2004) Building uniformly random subtrees. Random Struct. Alg. 24 420443.
[11] Lyons, R. (2005) Asymptotic enumeration of spanning trees. Combin. Probab. Comput. 14 491522.
[12] Lyons, R. (2010) Identities and inequalities for tree entropy. Combin. Probab. Comput. 19 303313.
[13] Lyons, R., Peled, R. and Schramm, O. (2008) Growth of the number of spanning trees of the Erdős–Rényi giant component. Combin. Probab. Comput. 17 711726.
[14] von Neumann, J. (1955) Mathematical Foundations of Quantum Mechanics (translated by Beyer, Robert T.), Princeton University Press.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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