A Cheeger type inequality in finite Cayley sum graphs

Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. 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 $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.


Introduction
Let G be a finite group, and S be a subset of G with |S| = d. The Cayley sum graph C Σ (G, S) is the graph having G as its set of vertices and for g, h ∈ G, the vertex h is adjacent to g if h = g −1 s for some element s ∈ S. These are classical combinatorial objects. We also recall that the Cayley graph of G (sometimes, called the Cayley difference graph), denoted by C(G, S), is the graph having G as its set of vertices and a vertex h is adjacent to a vertex g if h = gs for some element s ∈ S. The structures of C(G, S) and C Σ (G, S) can be very different. This can be seen by considering the Cayley graph C(G, S) and the Cayley sum graph C Σ (G, S) of G = Z/nZ (n 5) with respect to the set S = {±1}. The former is always a cycle graph while the latter need not be so (for instance, the latter is a cycle graph when n is even and is a path with loops at the endpoints whenever n is odd). Cayley graphs have been extensively studied over the ages. However, despite being classical combinatorial objects, the literature on Cayley sum graphs is not extensive. Very few things are known about Cayley sum graphs and most of them are quite recent works. They include those of Chung [7], Green [11,12], Green-Morris [13], Grynkiewicz-Lev-Serra [14], DeVos-Goddyn-Mohar-Šámal [9], Mrazović [18], Konyagin-Shkredov [15], Ma-Feng-Wang [17]. Indeed, even the question of vertex connectivity of abelian Cayley sum graphs was treated very recently in 2009 by Grynkiewicz-Lev-Serra [14], whereas vertex connectivity of abelian Cayley graphs is relatively easy. One reason for this fact is that unlike in the case of Cayley graphs, Cayley sum graphs may have less symmetry in them. Even less is known about the spectra of Cayley sum graphs. In fact, among the previous works, only that of Chung [7] and that of DeVos et. al. [9] deal with eigenvalues of some Cayley sum graphs, whereas the computation of distribution of eigenvalues of graphs is a fundamental topic of interest in graph theory. It is not yet known whether random Cayley sum graphs are expanders. Much remains to be discovered about Cayley sum graphs and it is a topic of current research. In this article, our main motivation is to establish a lower bound on the distribution of non-trivial eigenvalues of Cayley sum graphs.
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 multigraphs considered may admit loops. Given a finite d-regular multi-graph G = (V, E) where V denotes the set of vertices and E ⊆ V 2 denotes the multi-set of edges, we have the normalised adjacency matrix T of size |V | × |V |, which is equal to 1 d A, where A denote the adjacency matrix of G of size |V | × |V |, whose (i, j)-th entry is equal to the number of edges joining the i-th vertex and the j-th vertex of V .
The multi-graph G is connected if and only if λ 2 > 0 (equivalently, t 2 < 1). Moreover, if G is connected, then it is bipartite if and only if λ n = 2 (equivalently, t n = −1). For a subset V 1 ⊆ V , we denote the neighbourhood of V 1 by N (V 1 ) where, The boundary of V 1 is defined as δ(V 1 ) := N (V 1 )\V 1 .
Remark 1.1. In the case of a Cayley graph C(G, S), the neighbourhood N (V 1 ) of a subset V 1 of G is equal to V 1 S and its boundary δ(V 1 ) is equal to V 1 S S, while for a Cayley sum graph C Σ (G, S), the neighbourhood N (V 1 ) of a subset V 1 of G is equal to V −1 1 S, and its boundary δ(V 1 ) is equal to V −1 1 S S.
By considering a subset of V having size around half of the size of V , it follows that the vertex Cheeger constant of (V, E) cannot be much bigger than one. Indeed, if V contains an even number of elements, then taking a subset V 1 of V of size equal to |V |/2, one observes that the ratio of the size of its boundary δ(V 1 ) to the size of V 1 is bounded from the above by 1. Moreover, if V contains an odd number of elements, then taking a subset V 2 of V of size equal to (|V | − 1)/2, one observes that the ratio of the size of its boundary δ(V 2 ) to its size |V 2 | is bounded from the above by In particular, the vertex Cheeger constant of G satisfies h(G) 2. Next, we recall the notion of an expander graph as stated by Alon in [1].
. It was shown qualitatively by Breuillard, Green, Guralnick, and Tao that the eigenvalues of the normalised Laplacian matrix of non-bipartite finite Cayley graphs are bounded away from 2 [4, Appendix E]. Based on their arguments, the first author recently established an explicit upper bound [3,Theorem 1.4]. In this article, we show that a similar phenomenon occurs for the spectrum of the Cayley sum graph C Σ (G, S) by suitably adapting the strategy outlined in [4, Appendix E], along with the introduction of some new refinements. For an outline of the proof, we refer to § 1.1. Henceforth, we assume that |G| 4, to avoid trivial cases.
, where λ n (respectively, t n ) is the largest (respectively, the smallest) eigenvalue of the normalised Laplacian matrix (respectively, the normalised adjacency matrix) of C Σ (G, S).
Remark 1.5. Note that when C Σ (G, S) is bipartite, the spectrum of its adjacency matrix is symmetric about the origin. In this case, the lower spectrum is determined by the upper spectrum, for instance, 2d 2 , which follows from the discrete Cheeger-Buser inequality. Theorem 1.4 focuses on the non-bipartite Cayley sum graphs and shows that the smallest eigenvalue of the normalised adjacency matrix admits a lower bound depending only on the vertex Cheeger constant and the degree. This result is deduced after the proof of Theorem 2.10.
One has the following corollary of Theorem 1.4.
we have all the eigenvalues of the normalised adjacency matrix of each graph are uniformly bounded away from −1.
As a by-product of our proof, we improve the bound established for Cayley graphs in [3,Theorem 1.4]. See Theorem 2.12.
1.1. Outline of the proof. We outline the proof of Theorem 1.4. To prove this result, we assume on the contrary that the normalised adjacency matrix T of the Cayley sum graph admits an eigenvalue close to −1 (see Theorem 2.10). This implies that T 2 has an eigenvalue close to 1. We define a multi-graph M such that its normalised adjacency matrix is equal to T 2 (see the proof of Proposition 2.8). Then the discrete Cheeger-Buser inequality yields an upper bound on the edge-Cheeger constant of M, which in turn implies an upper bound on the vertex Cheeger constant of M. This yields a subset A of G of size |G| 2 having a convenient upper bound on |S −1 AS A|/|A|. Using combinatorial arguments, we obtain upper bounds on the sizes of several subsets defined using A (see Proposition 2.8). As a consequence, for a given element g ∈ G, we establish a dichotomy result on the size of A ∩ Ag (see Proposition 2.9), which states that the size A∩Ag is either very small or quite large as compared to the size of A. This allows us to adapt an argument due to Freȋman [10] in our set-up to construct a subgroup H + of G (see Theorem 2.10). From the bound on Algebraic Combinatorics, Vol. 4 #3 (2021) the smallest eigenvalue of T , it follows that the subgroup H + has index two in G. In Proposition 2.9, we also establish a similar dichotomy result on the size of A ∩ A −1 g. Using the strategy of Freȋman once again, we define a subset H − of G, which avoids S and is equal to a coset of H + in G, i.e. to H + or G H + . To conclude the result, we consider two cases. First, if H − is equal to H + , then the index two subgroup H + avoids S, which contradicts the hypothesis that C Σ (G, S) is non-bipartite (by Lemma 2.5). Next, if H − is equal to G H + , then the index two subgroup H + contains S, which contradicts the hypothesis that C Σ (G, S) is connected.

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 r-regular if each vertex has degree r. Apart from the vertex expansion as in Definition 1.3, we also have the notion of edge expansion.

Definition 2.2 (Edge-Cheeger constant). For a multi-graph G = (V, E), its edge-Cheeger constant h(G) is defined by
In a d-regular multi-graph, the two Cheeger constants are related by the following.

Lemma 2.3. Let G = (V, E) be a d-regular multi-graph. Then
holds.
Proof. Let ∅ = V 1 ⊆ V and we consider the map This map is surjective, hence we have the left hand side and at the worst case d to 1 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 [6] (the lower bound) and by Buser [5] (the upper bound). The discrete version was shown by Alon  Proof. The graph C Σ (G, S) is undirected if and only if for any g ∈ G and s ∈ S, (g −1 s) −1 t = g holds for some t ∈ S, which is equivalent to S being closed under conjugation. Proof which implies that ε > d − 1, which is impossible. Hence, the set B is empty. Then (2) follows from the inequality ε d, which holds by (1).
Algebraic Combinatorics, Vol. 4 #3 (2021) Proof. Since |G| 4, it follows from Proposition 2.7 that This implies that ζ < 1. Let T denote the normalised adjacency matrix of the Cayley sum graph C Σ (G, S). Since T has an eigenvalue in (−1, −1 + ζ] and ζ < 1, it follows that T 2 has an eigenvalue ν in [(1 − ζ) 2 , 1). Let M denote the weighted graph having G as its set of vertices and having T 2 as its normalised adjacency matrix. Thus the second largest eigenvalue of the normalised adjacency matrix of M is ν (1−ζ) 2 = 1 − ζ(2 − ζ). Hence the second smallest eigenvalue of the normalised Laplacian matrix of M is ζ(2 − ζ). By the discrete Cheeger-Buser inequality (Proposition 2.4), it follows that the edge-Cheeger constant of M satisfies

Consequently, by Lemma 2.3, the vertex Cheeger constant of M satisfies
This implies that for some non-empty subset A of G with |A| 1 2 |G|, holds. We claim that Otherwise, the inequality |A ∪ A −1 S| 1 2 |G| would imply which combined with the inequalities This contradicts the assumption ζ ε 2 4d 4 . Hence (5) holds. Applying Proposition 2.7 to the Cayley sum graph C Σ (G, S), we obtain where the last inequality follows from (4). This proves the inequalities as in statement (1).
To obtain the inequality in statement (2), note that |A| 1 2 |G| implies that |Ag ∩ This establishes the inequality in statement (2). To obtain the inequality in statement (3), it suffices to observe that holds, where the final inequality is obtained by applying statements (1) and (2).
To complete the proof, it suffices to observe that holds, where the final 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 it follows that for a given element g ∈ G, (1) exactly one of the inequalities