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Abstract
FAQs
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Introduction
Basic Results
References
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Mathematics
Pure Mathematics

A Survey on Spectra of infinite Graphs

Profile image of Bojan MoharBojan Mohar

1989, Bulletin of the London Mathematical Society

https://doi.org/10.1112/BLMS/21.3.209
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Abstract

Contents 1. Introduction 209 2. Linear operators associated with a graph 210 3. Basic results 211 4. Spectral radius, walk generating functions and spectral measures 214 5. Growth and isoperimetric number of a graph 219 6. Positive eigenfunctions 222 7. Graphs of groups, distance regular graphs and trees 223 8. Some remarks on applications in chemistry and physics 230

FAQs

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AI

How does adjacency spectrum for infinite graphs differ from finite graphs?add

The study shows that spectra of infinite graphs depend heavily on the choice of space; for instance, the adjacency operator acts differently on l^2 versus l^p spaces for infinite graphs.

What role does the growth function play in understanding graph properties?add

Growth functions, related to the transition operator, categorize graphs as exponential or polynomial growth; for example, a graph with exponential growth indicates that the probability of walks diverges rapidly.

When is the self-adjoint extension of an adjacency operator unique?add

The self-adjoint extension of the adjacency operator is unique if the deficiency index equals zero, which occurs when the degree of the graph is finite.

What implications does the isoperimetric number have on a graph's growth type?add

The research indicates that if the isoperimetric number i(G) is greater than zero, the graph has exponential growth; this relationship highlights critical connections between growth and structural properties.

How are transition operators characterized for locally finite infinite graphs?add

Transition operators for locally finite infinite graphs are self-adjoint with norms less than one, indicating behavior akin to stochastic processes with Markov properties.

Figures (3)
= Many authors—often independently—have contributed to spectral theory, functional analysis, walk generating functions, random walks, and so on, for the homogeneous (regular) tree 7, of degree q (Figure 2). (Sometimes, the homogeneous tree of degree g+1 is indexed with g because of its connection with g-adic groups [114].) If q is even, then 7, is the Cayley graph of a free group, and many papers deal with 7, in this ‘disguise’. The ‘classical ancestor’ is Kesten [76], who (among other things) calculates the closed walk generating function of the transition operator. This appears also in [48, 61], for example. The study of harmonic analysis for the adjacency (or, equivalently, the transition) operator of 7, goes back to [18] and has been developed by many authors. The results are subsumed in the book [45]; we point out the references [11, 24, 25, 27, 40, 44, 83, 110, 112]; further literature can be found in [45]. Some harmonic analysis of 7, can also be found in [81]. For the adjacency operator of 7, we have
= Many authors—often independently—have contributed to spectral theory, functional analysis, walk generating functions, random walks, and so on, for the homogeneous (regular) tree 7, of degree q (Figure 2). (Sometimes, the homogeneous tree of degree g+1 is indexed with g because of its connection with g-adic groups [114].) If q is even, then 7, is the Cayley graph of a free group, and many papers deal with 7, in this ‘disguise’. The ‘classical ancestor’ is Kesten [76], who (among other things) calculates the closed walk generating function of the transition operator. This appears also in [48, 61], for example. The study of harmonic analysis for the adjacency (or, equivalently, the transition) operator of 7, goes back to [18] and has been developed by many authors. The results are subsumed in the book [45]; we point out the references [11, 24, 25, 27, 40, 44, 83, 110, 112]; further literature can be found in [45]. Some harmonic analysis of 7, can also be found in [81]. For the adjacency operator of 7, we have
isoperimetric number i(T) > 0 (recall that this is equivalent with r(P(T)) < 1) if and only if there is an upper bound on the size of those induced subgraphs which are paths.
isoperimetric number i(T) > 0 (recall that this is equivalent with r(P(T)) < 1) if and only if there is an upper bound on the size of those induced subgraphs which are paths.

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