A Cheeger type inequality in finite Cayley sum graphs
Arindam Biswas, Jyoti Prakash Saha

TL;DR
This paper establishes a Cheeger-type inequality for finite Cayley sum graphs, providing explicit spectral bounds based on the Cheeger constant, and improves existing bounds on the spectrum of related Cayley graphs.
Contribution
It introduces a Cheeger-type inequality for Cayley sum graphs, linking spectral bounds to the Cheeger constant, and refines previous spectral bounds for Cayley graphs.
Findings
Spectral bounds depend on the Cheeger constant and degree.
Non-bipartite Cayley sum graphs have spectra bounded away from -1.
Improved bounds on the spectrum of non-bipartite Cayley graphs.
Abstract
Let be a finite group and be a symmetric generating set of with . We show that if the undirected Cayley sum graph is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from . We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval , where denotes the (vertex) Cheeger constant of the -regular graph and . Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph .
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A Cheeger type inequality in finite Cayley sum graphs
Arindam Biswas
Universität Wien, Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
and
Jyoti Prakash Saha
Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India
Abstract.
Let be a finite group and be a symmetric generating set of with . We show that if the undirected Cayley sum graph is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from . We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval , where denotes the (vertex) Cheeger constant of the -regular graph and . Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph .
Key words and phrases:
Expander graphs, Cheeger inequality, Spectra of Cayley sum graphs
2010 Mathematics Subject Classification:
05C25, 05C50, 05C75
1. Introduction
Let be a finite group, and be a symmetric generating set of not containing the identity element with . The Cayley sum graph is the graph having as its set of vertices and for , the vertex is adjacent to if for some element . These are classical combinatorial objects, e.g., see [GGL95] and [Gre17]. In this article, we consider the undirected Cayley sum graph and this is equivalent to saying that is closed under conjugation (see Lemma 2.6). We also recall that the Cayley graph of (sometimes called the Cayley difference graph) denoted by is the graph having as its set of vertices and the vertex is adjacent to if for some element . The structure of and can be very different. This can be seen considering the Cayley graph and the Cayley sum graph of () with respect to the symmetric generating set . The former is always a cycle graph while the latter need not be so (for instance, it admits loops whenever is odd).
In the following, the graphs and the multi-graphs considered are all undirected. The multi-graphs may possibly admit multiple edges. Moreover, the graphs and the multi-graphs considered may admit loops. Given a finite -regular multi-graph where denotes the set of vertices and the multi-set of edges, we have the normalised adjacency matrix of size whose eigenvalues lie in the interval . The normalised Laplacian matrix of is defined by
[TABLE]
where denotes the identity matrix of size . The eigenvalues of lie in the interval . Denote the eigenvalues of and the eigenvalues of as and respectively such that and
[TABLE]
The multi-graph is connected if and only if , while it is bipartite if and only if (equivalently ).
Let the multi-graph has vertex set and edge multi-set . For a subset , we denote the neighbourhood of as where,
[TABLE]
Then the boundary of is defined as .
Definition 1.1** (Vertex Cheeger constant).**
The vertex Cheeger constant of the multi-graph , denoted by , is defined as
[TABLE]
Next, we recall the notion of an expander graph as stated in [Alo86].
Definition 1.2** (-expander).**
Let . An -expander is a graph on vertices, having maximal degree , such that for every set satisfying , holds (equivalently,
We are interested in the spectrum of the expander graphs. It was remarked in [BGGT15] that if the eigenvalues of the normalised Laplacian matrix of non-bipartite finite Cayley graphs are bounded away from . Recently the first author established an explicit upper bound. See [Bis19, Theorem 1.4].
In this article, we show that a similar phenomenon occurs for the spectrum of the Cayley sum graph .
Theorem 1.3**.**
Let the Cayley sum graph be an expander with . Let denote its vertex Cheeger constant. Then if is non-bipartite, we have
[TABLE]
where (respectively ) is the largest (respectively smallest) eigenvalue of the normalised Laplacian matrix (respectively normalised adjacency matrix) of .
This result is deduced after the proof of Theorem 2.10. As a corollary of the above theorem it follows that
Corollary 1.4**.**
Let be an integer. Let be a sequence of non-bipartite, finite Cayley sum graphs with . Then, if there exists an uniform , such that each graph in the sequence is an -expander, we have all the eigenvalues of the normalised adjacency matrix of each graph are uniformly bounded away from .
As a by-product of our proof we improve the bound established for Cayley graphs in [Bis19, Theorem 1.4]. See Theorem 2.11. Further, we prove sharper estimates for both Cayley sum graphs and Cayley graphs under the assumption that no proper symmetric subset of generates . See Section 3, Theorem 3.2.
1.1. Outline of the proof
We outline the proof of Theorem 1.3. To prove this result, we assume on the contrary that the normalised adjacency matrix of the Cayley sum graph admits an eigenvalue close to (see Theorem 2.10). This implies that has an eigenvalue close to . We define a multi-graph such that its normalised adjacency matrix is equal to (see the proof of Proposition 2.8). Then the discrete Cheeger–Buser inequality yields an upper bound on the edge-Cheeger constant of , which in turn implies an upper bound on the vertex-Cheeger constant of . This yields a subset of of size having a convenient upper bound on . Using combinatorial arguments, we obtain upper bounds on the sizes of several subsets defined using (see Proposition 2.8). As a consequence, for a given element , we establish a dichotomy result on the size of (see Proposition 2.9), which states that the size is either very small or quite large as compared to the size of . This allows us to adapt an argument due to Freĭman [Fre73] in our set-up to construct a subgroup of (see Theorem 2.10). From the bound on the smallest eigenvalue of , it follows that the subgroup has index two in . In Proposition 2.9, we also establish a similar dichotomy result on the size of . Using the strategy of Freĭman once again, we define a subset of , which avoids and is equal to a coset of in , i.e., to or . To conclude the result, we consider two cases. First, if is equal to , then the index two subgroup avoids , which contradicts the hypothesis that is non-bipartite (by Lemma 2.5). Next, if is equal to , then the index two subgroup contains , which contradicts the hypothesis that generates .
1.2. Acknowledgements
We wish to thank Emmanuel Breuillard for a number of helpful discussions during the opening colloquium of the Münster Mathematics Cluster. The first author would like to acknowledge the support of the OWLF program and would also like to thank the Fakultät für Mathematik, Universität Wien where he was supported by the European Research Council (ERC) grant of Goulnara Arzhantseva, “ANALYTIC” grant agreement no. 259527. The second author would like to acknowledge the Initiation Grant from the Indian Institute of Science Education and Research Bhopal and the INSPIRE Faculty Award from the Department of Science and Technology, Government of India. He would also like to thank the MFO for their hospitality.
2. Proof of the main result
The degree of a vertex of a multi-graph is the number of half-edges adjacent to it (in the absence of loops). The presence of a loop at a vertex increases its degree by one. A multi-graph is said to be -regular if each vertex has degree . Apart from the vertex expansion as in Definition 1.2, we also have the notion of edge expansion.
Definition 2.1** (Edge expansion).**
Let be a -regular multi-graph with vertex set and edge multi-set . For a subset , let be the edge boundary of , defined as
[TABLE]
Then the edge expansion ratio of is defined as
[TABLE]
Definition 2.2** (Edge-Cheeger constant).**
The edge-Cheeger constant of a multi-graph is defined by
[TABLE]
In a -regular multi-graph the two Cheeger constants are related by the following -
Lemma 2.3**.**
Let be a -regular multi-graph
[TABLE]
Proof.
Let and we consider the map
[TABLE]
The map is surjective hence we have the left hand side and at the worst case to wherein we get the right hand side. ∎
The discrete Cheeger–Buser inequality relates the (edge) Cheeger constant with the second smallest eigenvalue of the Laplacian matrix. It is the version for graphs of the corresponding inequalities for the Laplace-Beltrami operator on compact Riemannian manifolds. It was first proven by Cheeger [Che70] (lower bound) and by Buser [Bus82] (upper bound). The discrete version was shown by Alon and Millman [AM85] (Proposition 2.4).
Proposition 2.4** (Discrete Cheeger–Buser inequality).**
Let be a finite -regular multi-graph. Let denote the second smallest eigenvalue of its normalised Laplacian matrix and be the (edge) Cheeger constant. Then
[TABLE]
Proof.
See [Lub94, Proposition 4.2.4, 4.2.5] or [Fri92, Section 1].
∎
Lemma 2.5**.**
The Cayley sum graph is bipartite if and only if contains a subgroup of index two which does not intersect .
Proof.
Suppose contains a subgroup of index two which does not intersect . Note that forms an independent subset of the set of vertices of the graph . Otherwise, for two adjacent elements with for some , we will obtain , which contradicts . We claim that also forms an independent subset of the set of vertices of the graph . Otherwise, for two adjacent elements with for some , we will obtain . Since has index two in , it follows that the product of any two elements of lying outside lies in . Thus we get . Hence is independent as claimed. So the Cayley sum graph is bipartite.
Suppose the Cayley sum graph is bipartite, i.e, its vertex set is the union of two disjoint partite sets . Without loss of generality, suppose contains the identity element of . Let be two elements of . Since is connected, the vertices are connected to . Since is symetric, the elements are equal to products of even number of elements of . So is also equal to a product of even number of elements of . Thus , and hence is a subgroup of . Since is independent, it does not intersect . Let be an element. Since is independent, the image of the map defined by does not intersect , and hence . Similarly, . So , and hence is a subgroup of of index two avoiding . ∎
Lemma 2.6**.**
The Cayley sum graph is undirected if and only if is closed under conjugation.
Proof.
Note that if is adjacent to , then , which implies that is adjacent to if and only if , i.e., is adjacent to each of its adjacent vertices if and only if . Hence is undirected if and only if closed under conjugation. ∎
Lemma 2.7**.**
Suppose is an -vertex expander for some , i.e.,
[TABLE]
for every subset with . Then for any subset of with , the inequality
[TABLE]
holds.
Proof.
The claimed inequality follows from
[TABLE]
and
[TABLE]
∎
Proposition 2.8**.**
Let be an -vertex expander for some . Suppose the normalised adjacency matrix of has an eigenvalue in the interval for some satisfying . Then for some subset of , the following conditions hold with .
- (1)
. 2. (2)
* for all .* 3. (3)
* for all .* 4. (4)
* for all .* 5. (5)
* for all .*
Proof.
Since is not bipartite, by Lemma 2.5, it follows that . Let be an element of . If has order , then and is of order , and hence
[TABLE]
When , we have
[TABLE]
which implies
[TABLE]
Consequently, it follows that . Let denote the normalised adjacency matrix of the Cayley sum graph . Since has an eigenvalue in and , it follows that has an eigenvalue in .
Consider the undirected multi-graph (which may contain multiple edges, also and multiple loops at a single vertex) with as its set of vertices and its edges are obtained by drawing an edge from to for each . Since is symmetric, this multi-graph is indeed undirected (since for any and for any ). For two distinct elements , the edges from to and are considered distinct (even when ). Note that the normalised adjacency matrix of is equal to . Thus the second largest eigenvalue of the normalised adjacency matrix of is . Hence the second smallest eigenvalue of the normalised Laplacian matrix of is . By the discrete Cheeger–Buser inequality (Proposition 2.4), it follows that the edge-Cheeger constant of satisfies
[TABLE]
which yields
[TABLE]
Consequently, by Lemma 2.3, the vertex-Cheeger constant of satisfies
[TABLE]
This implies that for some subset of with ,
[TABLE]
holds (since the size of the set is no larger than the size of the boundary of the subset of the set of vertices of ).
We claim that
[TABLE]
Otherwise, the inequality would imply
[TABLE]
which combined with the inequalities
[TABLE]
and
[TABLE]
implies
[TABLE]
This contradicts the assumption . Hence Equation (2.3) holds.
Applying Lemma 2.7 to the Cayley sum graph , we obtain
[TABLE]
So
[TABLE]
which implies
[TABLE]
where the last inequality follows from Equation (2.2). This proves the inequalities as in statement (1).
To obtain the inequality in statement (2), note that implies that . Since is an -vertex expander, it follows that
[TABLE]
This establishes the inequality in statement (2).
To obtain the inequality in statement (3), it suffices to observe that
[TABLE]
holds, where the strict inequality is obtained by applying statement (1) and (2).
To obtain the inequality in statement (4), note that implies that . Since is an -vertex expander, it follows that
[TABLE]
This establishes the inequality in statement (4).
To complete the proof, it suffices to observe that
[TABLE]
holds, where the strict inequality is obtained by applying statement (1) and (4). ∎
Proposition 2.9**.**
Under the notations and assumptions as in Proposition 2.8, and the additional hypothesis
[TABLE]
it follows that for a given element ,
- (1)
exactly one of the inequalities
[TABLE]
holds, 2. (2)
exactly one of the inequalities
[TABLE]
holds.
Proof.
Note that the inequalities
[TABLE]
imply that
[TABLE]
Hence it suffices to show that for a given element , one of the inequalities
[TABLE]
holds, and one of the inequalities
[TABLE]
holds.
Define the subset of by . The set is also equal to . Note that
[TABLE]
and
[TABLE]
hold as a consequence of Proposition 2.8(3). We consider the following cases, viz., . When holds, we obtain
[TABLE]
which yields
[TABLE]
Since
[TABLE]
holds, we obtain
[TABLE]
While holds, we obtain
[TABLE]
which yields
[TABLE]
Since
[TABLE]
holds, we obtain
[TABLE]
Considering the subset of defined by , and using Proposition 2.8(5) and similar arguments as above, we obtain that
[TABLE]
or
[TABLE]
holds according as or . ∎
Theorem 2.10**.**
Suppose is an -vertex expander for some . Assume that this graph is not bipartite. Then the eigenvalues of the normalised adjacency matrix of this graph are greater than with
[TABLE]
Proof.
On the contrary, let us assume that an eigenvalue of the normalised adjacency matrix of the graph lies in the interval . Since does not contain an index two subgroup by Lemma 2.5, it follows that is non-bipartite, and hence is not an eigenvalue of its normalised adjacency matrix. Hence an eigenvalue of the normalised adjacency matrix of the graph lies in the interval . Set
[TABLE]
Since , we have
[TABLE]
[TABLE]
Consequently,
[TABLE]
Define the subsets of by
[TABLE]
Note that contains the identity element of . By the triangle inequality,
[TABLE]
Consequently,
[TABLE]
If , then we obtain
[TABLE]
which implies . Since , by Proposition 2.9(1), it follows that contains . So is a subgroup of . Note that is not equal to , otherwise, we will obtain
[TABLE]
which yields .
The following estimate
[TABLE]
implies
[TABLE]
Using Proposition 2.8(1), we obtain
[TABLE]
We claim that is a subgroup of of index two. To prove this claim, it suffices to show that
[TABLE]
i.e.,
[TABLE]
which is equivalent to
[TABLE]
Let . Note that
[TABLE]
From Equation (2.6), it suffices to show that
[TABLE]
i.e., it suffices to show that
[TABLE]
Collecting the terms, it suffices to show that,
[TABLE]
which reduces to
[TABLE]
The above cubic polynomial in is positive for and hence the claim that is a subgroup of of index two follows.
By Proposition 2.8(2), does not intersect the set . Similar to as before, the following estimate
[TABLE]
implies
[TABLE]
This inequality combined with Proposition 2.8(1) yields
[TABLE]
The inequality in Equation (2.5) (which has been established) implies that
[TABLE]
and consequently, is nonempty. Note that for , the triangle inequality implies
[TABLE]
which yields
[TABLE]
If , then we will obtain
[TABLE]
which in turn implies . Since , using Proposition 2.9(2), we conclude that , i.e., contains . Thus, is contained in . Since is nonempty, it follows that is equal to or is equal to the non-trivial coset of in , i.e., . If is not equal to , then the index two subgroup of will contain (since ), which contradicts the fact that generates . So is equal to . Consequently, is a subgroup of of index two avoiding . Thus, the graph is bipartite by Lemma 2.5. We are done. ∎
Proof of Theorem 1.3.
Since is connected, its vertex Cheeger constant is positive. Thus is an -expander with . So Theorem 1.3 follows from Theorem 2.10. ∎
Proof of Corollary 1.4.
From Theorem 1.3, it follows that for any , the eigenvalues of the normalised adjacency matrix of of are greater than , which is depends on , but not on . Hence the corollary. ∎
As a consequence of the proof of Theorem 2.10, we obtain the following refinement of the bound provided in [Bis19, Theorem 1.4].
Theorem 2.11**.**
Let denote the Cayley graph of with respect to the symmetric generating set with . If this graph is non-bipartite and , then the largest eigenvalue of the normalised Laplacian matrix is less than
[TABLE]
Proof.
Suppose is an -vertex expander with and it is non-bipartite. We claim that the largest eigenvalue of the normalised Laplacian matrix is less than
[TABLE]
The bound on this eigenvalue given by [Bis19, Theorem 1.4] is
[TABLE]
Note that the proof of this result as in loc.cit. crucially relies on the last inequality in [Bis19, p.306], i.e., the inequality
[TABLE]
where . This inequality has been established using and the hypothesis that . The analogue of Equation (2.7) in the context of Cayley sum graph is the inequality
[TABLE]
in Equation (2.6) where . The above inequality has been established using and . Hence Equation (2.7) will follow for if holds, which is true by Lemma 2.12 below. So the claim follows. Noting that is an -vertex expander, and (since the graph is connected), the result follows from the claim. ∎
Lemma 2.12**.**
**
Proof.
Since is an -expander,
[TABLE]
Let and contains an element such that . Let . Then
[TABLE]
If and all elements of have order and then choose for some . Proceeding as above, it is clear in this case that or . In the remaining cases, the inequality follows by a case by case analysis on the size of . This proves the Lemma. ∎
3. Sharper estimates
Lemma 3.1**.**
Suppose the Cayley sum graph is non-bipartite and no symmetric set satisfying generates . If is -vertex expander with , then .
Proof.
Note that contains at least two elements. Otherwise, it contains only one element, and it is of order two (since is symmetric), in which case is bipartite by Lemma 2.5. If contains only two elements, then .
Suppose contains at least three elements. Let be an element of . Note that the is a nonempty symmetric subset of . Let denote the subgroup of generated by the . Since , we obtain
[TABLE]
which yields . ∎
Theorem 3.2**.**
Suppose is an -vertex expander for some . Assume that this graph is not bipartite, and no symmetric set satisfying generates . Set
[TABLE]
If , then the eigenvalues of the normalised adjacency matrix of this graph are greater than whenever and take the values as in Table 1.
Proof.
Note that the proof of Theorem 2.10 depends on through Equation (2.4) and (2.6). Hence it suffices to prove that these two equations hold for the redefined as in Equation (3.1). If , then the inequality
[TABLE]
implies that Equation (2.4) holds. By Lemma 3.1, we obtain . Using this estimate, it turns out that
[TABLE]
is less than , i.e., the inequality in Equation (2.6) holds whenever , and and take the prescribed values. Hence the conclusion of Theorem 2.10 holds when is redefined as above, and and satisfy the given conditions. ∎
Note that Lemma 3.1 holds when is replaced by . Hence Theorem 3.2 remains valid even when the Cayley sum graph is replaced by the Cayley graph .
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[Alo 86] N. Alon, Eigenvalues and expanders , Combinatorica 6 (1986), no. 2, 83–96, Theory of computing (Singer Island, Fla., 1984). MR 875835
- 2[AM 85] N. Alon and V. D. Milman, λ 1 , subscript 𝜆 1 \lambda_{1}, isoperimetric inequalities for graphs, and superconcentrators , J. Combin. Theory Ser. B 38 (1985), no. 1, 73–88. MR 782626
- 3[BGGT 15] Emmanuel Breuillard, Ben Green, Robert Guralnick, and Terence Tao, Expansion in finite simple groups of Lie type , J. Eur. Math. Soc. (JEMS) 17 (2015), no. 6, 1367–1434. MR 3353804
- 4[Bis 19] Arindam Biswas, On a Cheeger type inequality in Cayley graphs of finite groups , European J. Combin. 81 (2019), 298–308. MR 3975766
- 5[Bus 82] Peter Buser, A note on the isoperimetric constant , Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
- 6[Che 70] Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian , Princeton Univ. Press, Princeton, N. J., 1970. MR 0402831
- 7[Fre 73] G. A. Freĭman, Groups and the inverse problems of additive number theory , Number-theoretic studies in the Markov spectrum and in the structural theory of set addition (Russian), Kalinin. Gos. Univ., Moscow, 1973, pp. 175–183. MR 0435006
- 8[Fri 92] Shmuel Friedland, Lower bounds for the first eigenvalue of certain M 𝑀 M -matrices associated with graphs , Linear Algebra Appl. 172 (1992), 71–84, Second NIU Conference on Linear Algebra, Numerical Linear Algebra and Applications (De Kalb, IL, 1991). MR 1168497
