Note on the product of the largest and the smallest eigenvalue of a graph

In this note, we use eigenvalue interlacing to derive an inequality between the maximum degree of a graph and its maximum and minimum adjacency eigenvalues. The case of equality is fully characterized.


Introduction
Spectral graph theory seeks to deduce the structural properties of a graph from its spectrum.Eigenvalue interlacing provides a powerful tool for obtaining inequalities and regularity results concerning the structure of graphs in terms of eigenvalues of the adjacency matrix; see, for instance, Brouwer and Haemers [4] and Haemers [10].In addition, studying the cases of equality in such inequalities often provides interesting information on the structure of the graph.
Let G be a (connected) graph with n vertices and adjacency matrix having eigenvalues λ 1 ≥ • • • ≥ λ n .Not many bounds involving the product of the largest and smallest eigenvalue, λ 1 and λ n , of a graph seem to exist.In fact, we are only aware of an inequality obtained by Haemers [10,Theorem 3.3], who provides an upper bound for the independence number α of a non-regular graph, extending the celebrated Hoffman ratio bound for regular graphs α ≤ n/[1−(λ 1 /λ n )] (see [10,Theorem 3.2]).Namely, Haemers proved that, if G is a graph with n vertices, minimum degree δ, and independence number α, then Several papers, such as the one by Gregory, Hershkowitz, and Kirkland [9], are on the graph spread, that is, λ 1 − λ n .There is also quite some work on bounding the sum λ 1 + λ n , see, for instance, Brandt [3], Csikvári [5] and Balogh, Clemen, Lidický, Norin, and Volec [2].Moreover, bounds on λ 1 knowing λ n , or the other way around, have been obtained by Rojo, Rojo, and Soto [11].
The structure of this note is as follows.We start Section 2 by providing some basic definitions and some known results, such as the interlacing theorem.In Section 3, we present our main result, that is, an inequality involving only the maximum degree ∆ of a graph and the product of its maximum and minimum adjacency eigenvalues.Moreover, the case of equality is fully characterized, and some infinite families satisfying it are provided.Finally, in Section 4 we show that our bound can outperform the other known bound of a similar nature by Haemers [10,Theorem 3.3].

Preliminaries
Throughout this note, G = (V, E) denotes a (simple and connected) graph with n = |V | vertices, and adjacency matrix A with eigenvalues Moreover, G(u) stands for the set of vertices adjacent to a given vertex u ∈ V .
Given two square matrices A and B with eigenvalues with m < n), we say that the second sequence interlaces the first if, for all i = 1, . . ., m, it follows that λ i ≥ µ i ≥ λ n−m+i .
A basic result about interlacing is the following (see Haemers [10]).
Theorem 1.Let S be a real n × m matrix such that S T S = I, and let A be an n × n matrix with eigenvalues λ 1 ≥ • • • ≥ λ n .Define B = S T AS, and call its eigenvalues Let A be the adjacency matrix of a graph G = (V, E).We distinguish two interesting cases of eigenvalue interlacing depending on the matrix B: 1.If B is a principal submatrix of A, then B corresponds to the adjacency matrix of an induced subgraph G ′ of G.

If for a given partition of the vertices of
Notice that λ 1 (> δ) is the spectral radius of G, whereas, in principle, λ 2 is not necessarily the minimum eigenvalue of G.

The new bound
In this section we present our main result: an upper bound on the product of the maximum and minimum eigenvalues of a graph.This bound is very useful when we only know information about the maximum degree of the graph.
Theorem 3. Let G be a graph on n vertices with maximum degree ∆ = ∆(G) and eigenvalues Moreover, equality holds if and only if G = H ′ + u is a cone graph over a δ-regular graph H ′ on ∆ = n − 1 vertices (the degree of u) and minimum eigenvalue satisfying Proof.Let u be a vertex of G with maximum degree ∆, and consider the graph H induced by the vertex set {u} ∪ G(u).Let us first prove that H satisfies (2).If A ′ is the adjacency matrix of the graph H ′ = G(u) induced by G(u), the adjacency matrix A of H is of the form where j is the all-1 vector with ∆ entries.Then, its quotient matrix (where each block is replaced by the average sum of its arrows) is where δ is the mean degree of H ′ .Since the eigenvalues of we have, by interlacing, Then, noting that we have and ( 2) follows.In fact, as pointed out by Haemers (personal communication, 2024), this can be proved more directly by noting that Now, since H is an induced subgraph of G, we have, again by interlacing, Thus, using (6), and ( 2) holds.Now, the equality in (2) first implies equalities in (6), that is λ max (H) = λ 1 and λ min (H) = λ 2 .Consequently, we have tight interlacing, and (4) corresponds to a regular partition of H.That is, H = H ′ + v is a cone graph over the regular graph H ′ with degree δ = δ.Second, assuming that H is a proper subgraph of G, the equalities in (7) imply that λ max (G) = λ max (H).However, this is impossible since the spectral radius of a proper subgraph of a graph G is known to be smaller than the spectral radius of G (see, for instance, Cvetković, Rowlinson and Simić [6,Prop. 3.1.10]).Therefore, 4∆ and, hence, we get the equality in (2).
For example, (3) is satisfied when G is a cone graph of the following type: , and let us check that the condition (3) on H ′ holds.(Recall that ∆ is used both for the degree of u, or maximum degree of G, and the number of vertices of H ′ .) and its smallest eigenvalue is λ min (H ′ ) = −1 = φ(n − 2, n − 1).
(ii) The graph H ′ = (n − 1)K 1 with ∆ = n − 1 isolated vertices has degree δ = 0, and its smallest eigenvalue is From Theorem 3, we obtain the following straightforward consequences.

Bounds comparison
When equality does not hold in (2), it is interesting to compare our bound to the one by Haemers (1).Apart from the fact that (2) is more 'economical' in the sense that it uses only the maximum degree, in general, (2) gives a better bound for −λ 1 λ n when α is not very large.Namely, from ∆ ≥ αδ 2 /(n − α), we obtain that (2) outperforms (1) when In contrast, Haemers' bound (1) is better than (2) for the case of bipartite biregular graphs.In fact, we will show that, for such graphs, equality holds in (1).A bipartite graph G = (V, E) with V = V 1 ∪ V 2 is biregular if all the vertices of the stable set V 1 have degree k 1 , whereas all the vertices of V 2 have degree k 2 .Then, notice that by counting in two ways the number of edges of G, we have

(
The following result is well known; see, for instance, Brouwer and Haemers [4, Ch. 1, Ex. 6].Lemma 2. Let G = H ′ + v be the cone graph on n vertices, over the δregular (connected) graph H ′ with n − 1 vertices and eigenvalues θ 1 The entries of B, which are denoted by b ij for i, j = 1, ..., m, are the average row sums of the corresponding blocks A ij of A.Moreover, if the interlacing is tight, Theorem 1(iii) reflects that S corresponds to a regular (or equitable) partition of A, that is, each block of the partition has constant row and column sums.Then, the bipartite induced subgraphs G ij , with adjacency matrices A ij , for i = j, are biregular, and the subgraphs G ii are regular.The cone graph over a graph H is obtained by adding a new vertex v and joining it to all vertices of H.

Table 1 :
Proportion of small irregular graphs for which the new bound (2) outperforms the known bound (1).
(i) If G is a (non-trivial) regular graph, then its minimum eigenvalue satisfies λ min ≤ −1, with equality if and only if G is a complete graph.(ii)If G is a bipartite graph with maximum degree ∆ and spectral radius λ max , then ∆ ≤ λ 2 max , and equality holds if and only if G = S ∆ .