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PACIFIC JOURNAL OF MATHEMATICS 2022年第2期317卷 423-440页

作者： Oh, Josiah Pengitore, Mark Ohio State Univ Columbus OH 43210 USA Univ Virginia Charlottesville VA USA

We study nontransitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group G, we produce continuum many pairwise non-quasiisometric regular ... 详细信息

关键词： transitive graph quasiisometry growth rate number of ends asymptotic dimension

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Quasi-Isometric Rigidity of a Product of Lattices, and Coarse Geometry of Non-transitive graphs

Quasi-Isometric Rigidity of a Product of Lattices, and Coars...

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作者： Oh, Josiah The Ohio State University

学位级别：Ph.D., Doctor of Philosophy

In this dissertation we accomplish two things. 1. We demonstrate quasi-isometric rigidity for the product L × Λ, where L is a lattice in a simply connected nilpotent Lie group and Λ is a non-uniform lattice in the isometry group of a negatively-curved symmetric space. We show that if Γ is a finitely generated group quasi-isometric to L × Λ, then up to finite noise, Γ is an extension of a non-uniform rank one lattice by a lattice in a simply connected nilpotent Lie group. Under additional hypotheses we further show that this extension is nilcentral, a notion which generalizes central extensions to extensions by a nilpotent group. 2. We study the large-scale geometry of non-transitive graphs. In particular we introduce a geometric generalization of vertex-transitivity for graphs, called coarse transitivity, and we prove that the classical Freudenthal-Hopf theorems on the ends of finitely generated groups hold more generally for coarsely transitive graphs. We also prove that for each finitely generated group G, there exist uncountably many regular graphs, no two of which are quasi-isometric, that share several geometric properties with G.

关键词： Geometric group theory Quasi-isometry Quasi-isometric rigidity Lattice Rank one Lie group Simply connected nilpotent Lie group transitive graph Growth rate Number of ends Asymptotic dimension

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Factor-of-iid Schreier decorations of lattices in Euclidean spaces

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DISCRETE MATHEMATICS 2024年第9期347卷

作者： Bencs, Ferenc Hruskova, Aranka Toth, Laszlo Marton Univ Amsterdam Amsterdam Netherlands Cent European Univ Dept Math & Its Applicat Budapest Hungary Alfred Renyi Inst Math HUN REN Budapest Hungary

A Schreier decoration is a combinatorial coding of an action of the free group Fd on the vertex set of a 2d-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor of iid. We show that Zd, d >= 3, the square lattice and also the three other Archimedean lattices of even degree have finitaryfactor-of-iid Schreier decorations, and, answering a question of Thornton, exhibit examples of transitive graphs of arbitrary even degree in which obtaining such a decoration as a factor of iid is impossible. We also prove that symmetrical planar lattices with all degrees even have a factor-of-iid balanced orientation, meaning the indegree of every vertex is equal to its outdegree, and demonstrate the existence of transitive graphs G whose classical chromatic number chi (G) is equal to their factor-of-iid chromatic number.

关键词： transitive graph Factor of iid Schreier graph Site percolation Archimedean lattice Planar lattice

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SELF-AVOIDING WALK ON NONUNIMODULAR transitive graphS

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ANNALS OF PROBABILITY 2019年第5期47卷 2801-2829页

作者： Hutchcroft, Tom Univ Cambridge DPMMS Statslab Cambridge CB3 0WA England

We study self-avoiding walk on graphs whose automorphism group has a transitive nonunimodular subgroup. We prove that self-avoiding walk is ballistic, that the bubble diagram converges at criticality, and that the critical two-point function decays exponentially in the distance from the origin. This implies that the critical exponent governing the susceptibility takes its mean-field value, and hence that the number of self-avoiding walks of length n is comparable to the nth power of the connective constant. We also prove that the same results hold for a large class of repulsive walk models with a self-intersection based interaction, including the weakly self-avoiding walk. All of these results apply in particular to the product T-k x Z(d) of a k-regular tree (k >= 3) with Z(d), for which these results were previously only known for large k.

关键词： Self-avoiding walk nonunimodular transitive graph mean-field nonamenable bubble diagram

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Topological indices of the wreath product of graphs

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DISCRETE MATHEMATICS ALGORITHMS AND APPLICATIONS 2023年第1期15卷

作者： Zhang, Yongqin Wang, Jianfeng Brunetti, Maurizio Shandong Univ Technol Sch Math & Stat Zibo 255049 Peoples R China Univ Naples Federico II Dept Math & Applicat Naples Italy

Topological indices, i.e., numerical invariants suitably associated to graphs and only depending upon their isomorphism type, have important applications in Chemistry. Their computation constitutes an important branch of Chemical graph Theory. In this paper, we focus on some degree and distance-based invariants related to the Zagreb indices, the Szeged index and the Wiener index, namely, the F-index, the vertex PI index and the hyper-Wiener index. In particular, we find the formula to compute these topological invariants for wreath product of graphs.

关键词： Wreath product forgotten index PI index hyper-Wiener index transitive graph

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Weighted self-avoiding walks

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JOURNAL OF ALGEBRAIC COMBINATORICS 2020年第1期52卷 77-102页

作者： Grimmett, Geoffrey R. Li, Zhongyang Univ Cambridge Ctr Math Sci Stat Lab Wilberforce Rd Cambridge CB3 0WB England Univ Melbourne Sch Math & Stat Parkville Vic 3010 Australia Univ Connecticut Dept Math Storrs CT 06269 USA

We study the connective constants of weighted self-avoiding walks (SAWs) on infinite graphs and groups. The main focus is upon weighted SAWs on finitely generated, virtually indicable groups. Such groups possess so-called height functions, and this permits the study of SAWs with the special property of being bridges. The group structure is relevant in the interaction between the height function and the weight function. The main difficulties arise when the support of the weight function is unbounded, since the corresponding graph is no longer locally finite. There are two principal results, of which the first is a condition under which the weighted connective constant and the weighted bridge constant are equal. When the weight function has unbounded support, we work with a generalized notion of the 'length' of a walk, which is subject to a certain condition. In the second main result, the above equality is used to prove a continuity theorem for connective constants on the space of weight functions endowed with a suitable distance function.

关键词： Self-avoiding walk Weight function Connective constant Bridge constant Virtually indicable group Cayley graph transitive graph

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Cubic graphs and the golden mean

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DISCRETE MATHEMATICS 2020年第I期343卷 111638-000页

The connective constant mu(G) of a graph G is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We investigate the validity of the inequality mu >= phi for infinite, transitive, simple, cubic graphs, where phi :=1/2(1 + root 5) is the golden mean. The inequality is proved for several families of graphs including (i) Cayley graphs of infinite groups with three generators and strictly positive first Betti number, (ii) infinite, transitive, topologically locally finite (TLF) planar, cubic graphs, and (iii) cubic Cayley graphs with two ends. Bounds for mu are presented for transitive cubic graphs with girth either 3 or 4, and for certain quasi-transitive cubic graphs. Published by Elsevier B.V.

关键词： Self-avoiding walk Connective constant Cubic graph transitive graph Grigorchuk group TLF-planar graph

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Full algebras of matrices

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LINEAR & MULTILINEAR ALGEBRA 2019年第8期67卷 1511-1521页

作者： Cigler, Grega Jerman, Marjan Wojciechowski, Piotr J. Univ Ljubljana Fac Math & Phys Ljubljana Slovenia Univ Texas El Paso Dept Math Sci El Paso TX 79968 USA

A classification of full algebras of matrices is given. All such algebras are permutation-isomorphic to block lower-triangular matrices with corresponding subdiagonal blocks being either zero-blocks or full. Two full algebras are isomorphic if and only if they are permutation-isomorphic. A one-to-one correspondence is provided between the full algebras and transitive directed graphs. It is also proven that such algebras, if endowed with a lattice order, can be almost f- or d-algebras only if they are diagonal.

关键词： Algebra of matrices transitive graph lattice order

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Self-Avoiding Walks and Connective Constants 1

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Conference on Probability Theory and Statistical Physics

作者： Grimmett, Geoffrey R. Li, Zhongyang Univ Cambridge Ctr Math Sci Stat Lab Wilberforce Rd Cambridge CB3 0WB England Univ Connecticut Dept Math 341 Mansfield Rd U1009 Storrs CT 06269 USA

ISBN: (数字)9789811503023

ISBN: (纸本)9789811503023;9789811503016

The connective constant mu(G) of a quasi-transitive graph G is the asymptotic growth rate of the number of self-avoiding walks (SAWs) on G from a given starting vertex. We survey several aspects of the relationship between the connective constant and the underlying graph G. - We present upper and lower bounds for mu in terms of the vertex-degree and girth of a transitive graph. - We discuss the question of whether mu >= phi for transitive cubic graphs (where phi denotes the golden mean), and we introduce the Fisher transformation for SAWs (that is, the replacement of vertices by triangles). - We present strict inequalities for the connective constants mu(G) of transitive graphs G, as G varies. - As a consequence of the last, the connective constant of a Cayley graph of a finitely generated group decreases strictly when a new relator is added, and increases strictly when a non-trivial group element is declared to be a further generator. - We describe so-called graph height functions within an account of `bridges' for quasi-transitive graphs, and indicate that the bridge constant equals the connective constant when the graph has a unimodular graph height function. - A partial answer is given to the question of the locality of connective constants, based around the existence of unimodular graph height functions. - Examples are presented of Cayley graphs of finitely presented groups that possess graph height functions (that are, in addition, harmonic and unimodular), and that do not. - The review closes with a brief account of the 'speed' of SAW.

关键词： Self-avoiding walk Connective constant Regular graph transitive graph Quasi-transitive graph Cubic graph Golden mean Fisher transformation Cayley graph Bridge constant Locality theorem graph height function Unimodularity Speed

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A conditional edge connectivity of double-orbit networks

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FUTURE GENERATION COMPUTER SYSTEMS-THE INTERNATIONAL JOURNAL OF ESCIENCE 2018年 83卷 445-449页

作者： Lin, Huiqiu Yang, Weihua East China Univ Sci & Technol Sch Sci Dept Math Shanghai 200237 Peoples R China Taiyuan Univ Technol Dept Math Taiyuan 030024 Shanxi Peoples R China

The edge connectivity is a kind of classic measure of fault tolerance of networks. It is well known that the edge-connectivity of a simple, connected, vertex transitive graph attains its regular degree. It is then natural to consider the relationship between the edge connectivity and the number of orbits of its automorphism group. The double-orbit graphs with two orbits of the same size is a generalization of vertex transitive networks, which contains several classic network models. In this note, we obtain a sufficient condition for such double-orbit graphs to be super-lambda'. (C) 2017 Elsevier B.V. All rights reserved.

关键词： graphs and networks Edge-connectivity Orbit Super-edge-connected transitive graph

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