A lower bound on the spectral radius of the universal cover of a graph
01 Jan 2005-Journal of Combinatorial Theory, Series B (Academic Press, Inc.)-Vol. 93, Iss: 1, pp 33-43
TL;DR: It is proved that if the average degree of the graph G after deleting any radius r ≥ 2 ball is at least d ≥ 2, then its second largest eigenvalue in absolute value λ(G) is at at least 2 √d - 1(1 - c log r/r).
About: This article is published in Journal of Combinatorial Theory, Series B.The article was published on 2005-01-01 and is currently open access. It has received 34 citations till now. The article focuses on the topics: Bound graph & Graph power.
Citations
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TL;DR: In this paper, a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) is presented.
Abstract: Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These
9,929 citations
TL;DR: Expander graphs were first defined by Bassalygo and Pinsker in the early 1970s, and their existence was proved in the late 1970s as discussed by the authors and early 1980s.
Abstract: A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even
2,037 citations
Journal Article•
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TL;DR: In this article, the authors define small submodules of a module M over R over a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E.
Abstract: The concept of a continuous module is a generalization of that of an injective module, and conditions (), (C) and () are given for this concept in [4]. In this paper, we study modules with properties that are dual to continuity. These will be called discrete and we discuss discrete abelian groups. Throughout R is a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E. Dual to the notion of essential submodules, we define small submodules of a module M over R.(omitted)
235 citations
Posted Content•
TL;DR: In this paper, it was shown that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly Θ(n), where n is the growth rate of the tree.
Abstract: A non-backtracking walk on a graph, $H$, is a directed path of directed edges of $H$ such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency matrix, $B$, indexed by $H$'s directed edges and related to Ihara's Zeta function. We show how to determine $B$'s spectrum in the case where $H$ is a tree covering a finite graph. We show that when $H$ is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of $B$'s spectrum, the corresponding Green function has ``periodic decay ratios.'' The existence of such a ``ratio system'' can be effectively checked, and is equivalent to being outside the spectrum. We also prove that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly $\sqrt\gr$, where $\gr$ is the growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras \cite{ST}. Finally, we give experimental evidence that for a fixed, finite graph, $H$, a random lift of large degree has non-backtracking new spectrum near that of $H$'s universal cover. This suggests a new generalization of Alon's second eigenvalue conjecture.
81 citations
Cites background or methods from "A lower bound on the spectral radiu..."
...[HLW] S. Hoory, N. Linial, and A. Wigderson....
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...A third motivation is the result of Alon, Hoory, and Linial [AHL02], giving a lower bound on the number of vertices in a graph with a specified girth and average degree....
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...(Omer Angel) Math, UofT E-mail address : angel@math.utoronto.ca (Joel Friedman) CS, UBC E-mail address : jf@cs.ubc.ca (Shlomo Hoory) Haifa Research Lab., IBM E-mail address : shlomoh@il.ibm.com u v ∞ ∞ ∞0 0 0 u v ∞ ∞ 0 0 u v ∞ ∞0 0...
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...In further work, Hoory [Hoo05], gave a similar bound on the spectral radius of H ’s universal cover, ρ(A(H̃))....
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...[Hoo05] S. Hoory....
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TL;DR: In this article, it was shown that the spectral radius of the non-backtracking walk operator on a tree covering a finite graph is exactly √ gr, where gr is the growth rate of the tree.
Abstract: A non-backtracking walk on a graph, H, is a directed path of directed edges of H such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency matrix, B, indexed by H's directed edges and related to Ihara's Zeta function. We show how to determine B's spectrum in the case where H is a tree covering a finite graph. We show that when H is not regular, this spectrum can have positive measure in the complex plane, unlike the regular case. We show that outside of B's spectrum, the corresponding Green function has "periodic decay ratios." The existence of such a "ratio system" can be effectively checked, and is equivalent to being outside the spectrum. We also prove that the spectral radius of the non-backtracking walk operator on the tree covering a finite graph is exactly √ gr, where gr is the growth rate of the tree. This further motivates the definition of the graph theoretical Riemann hypothesis proposed by Stark and Terras (ST). Finally, we give experimental evidence that for a fixed, finite graph, H, a ran- dom lift of large degree has non-backtracking new spectrum near that of H's universal cover. This suggests a new generalization of Alon's second eigenvalue conjecture.
72 citations
References
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TL;DR: In this paper, a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) is presented.
Abstract: Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These
9,929 citations
"A lower bound on the spectral radiu..." refers background in this paper
...A theorem due to Greenb erg [4] states that if $ is a family of graphs of size+ , , covered by the same universal cover ....
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Book•
01 Aug 1994
TL;DR: The Banach-Ruziewicz Problem for n = 2, 3 Ramanujan Graphs is solved and the representation theory of PGL 2 is explained.
Abstract: Expanding Graphs.- The Banach-Ruziewicz Problem.- Kazhdan Property (T) and its Applications.- The Laplacian and its Eigenvalues.- The Representation Theory of PGL 2.- Spectral Decomposition of L 2(G(?)\G(A)).- Banach-Ruziewicz Problem for n = 2, 3 Ramanujan Graphs.- Some More Discrete Mathematics.- Distributing Points on the Sphere.- Open Problems.
741 citations
TL;DR: For any prime power q, explicit constructions for many infinite linear families of q + 1 regular Ramanujan graphs are given as Cayley graphs of PGL2 or PSL2 over finite fields, with respect to very simple generators.
361 citations
"A lower bound on the spectral radiu..." refers background in this paper
...There are known constructions [7, 8, 9] for infinite families of*-regular Ramanujan graphs for every * that is a prime power plus one....
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TL;DR: In this paper, it was shown that the second largest eigenvalue of the adjacency matrix of any d-regular graph G containing two edges is at least 2 d − 1 − (2 d−1 − 1)/(k+1).
348 citations
"A lower bound on the spectral radiu..." refers background in this paper
.../0$12 $ 34* 5 6 (3) Note that this is indeed a generalization of the Alon Boppanabound for regular graphs, because for any sequence of*-regular graphs $ with diameter+ , , we can define monotone functions* + * and + , so that $ has an -robust average degree* ....
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...It turns out that there is an analogue for Alon Boppana also for non-regular graphs....
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...For regular graphs ) *, and a well known result of Alon and Boppana ([11] or [2]) states hat for any sequence of*-regular graphs $ of diameter+ , , -.! ....
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...We prove that for any graph of average degree and derive from it the following generalization of the Alon Boppana bound....
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...For regular graphs ) *, and a well known result of Alon and Boppana ([11] or [2]) state s hat for any sequence of *-regular graphs $ of diameter+ , , -....
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Journal Article•
[...]
TL;DR: In this article, the authors define small submodules of a module M over R over a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E.
Abstract: The concept of a continuous module is a generalization of that of an injective module, and conditions (), (C) and () are given for this concept in [4]. In this paper, we study modules with properties that are dual to continuity. These will be called discrete and we discuss discrete abelian groups. Throughout R is a ring with identity, M is a module over R, G is an abelian group of finite rank, E is the ring of endomorphisms of G and S is the center of E. Dual to the notion of essential submodules, we define small submodules of a module M over R.(omitted)
235 citations
"A lower bound on the spectral radiu..." refers background in this paper
...It is well known (see [6] chapter 4), that for any vertex , ) -....
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