# Extremal Graphs for a Spectral Inequality on Edge-Disjoint Spanning Trees

Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph G with minimum degree δ≥ 2m+2 ≥ 4 satisfies λ_2(G) < δ - 2m+1/δ+1, then G contains at least m+1 edge-disjoint spanning trees, which verified a generalization of a conjecture by Cioabă and Wong. We show this bound is essentially the best possible by constructing d-regular graphs 𝒢_m,d for all d ≥ 2m+2 ≥ 4 with at most m edge-disjoint spanning trees and λ_2(𝒢_m,d) < d-2m+1/d+3. As a corollary, we show that a spectral inequality on graph rigidity by Cioabă, Dewar, and Gu is essentially tight.

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## 1. Introduction

Let be a finite, simple graph on vertices, and let and be the eigenvalues of its adjacency and Laplacian matrices, respectively. Recall for -regular graphs [brouwer2011spectra, Ch. 1]. Additionally, let denote the maximum number of edge-disjoint spanning trees in , sometimes referred to as the spanning tree packing number (see Palmer [spanningpackingpalmer] for a survey of this parameter). Motivated by Kirchhoff’s celebrated matrix tree theorem on the number of spanning trees of a graph [kirchhoff1847ueber] and a question of Seymour [seymour], Cioabă and Wong [cioabaSpanningTrees] considered the relationship between the eigenvalues of a regular graph and .

They obtained a result by combining two useful theorems. The Nash-Williams/Tutte theorem [nash1961edge, tutte1961problem] (described in Section 2.2) implies that if is a -edge-connected graph, then . Additionally, Cioabă [cioabua2010eigenvalues] showed if is a -regular graph and is an integer with such that , then is -edge-connected. These facts imply that if is a -regular graph with for some integer , with , then contains edge-disjoint spanning trees. Cioabă and Wong conjectured the following factor of two improvement, which they verified for .

###### Conjecture 1 ([cioabaSpanningTrees]).

Let be an integer and be a -regular graph with . If , then .

This conjecture attracted much attention, leading to many partial results and generalizations [guthesis, spanningpacking, gu2016edge, hong2016fractional, li2013edge, liu2014edge]. The question was ultimately resolved by Liu, Hong, Gu, and Lai.

###### Theorem 2 ([noteOnTrees2014]).

Let be an integer and be a graph with minimum degree . If , then .

We show this bound is essentially the best possible.

###### Theorem 3.

For all , the -regular graph (defined in Section 2.1) has at most edge-disjoint spanning trees and satisfies

 d−2m+1d+1≤λ2(Gm,d)

Cioabă and Wong created special cases of this construction for the families and (a slight variant of) in [cioabaSpanningTrees] to show that Theorem 2 is essentially best possible for . In his PhD thesis [wongthesis], Wong also constructed the family to show that Theorem 2 is essentially tight for . Based on the family of graphs for the small cases of that appeared in [cioabaSpanningTrees], Gu [spanningpacking] constructed a family of multigraphs by replacing every edge with multiple edges to show that the bounds in a multigraph analog of Theorem 2 are also the best possible. Additionally, Cioabă, Dewar, and Gu [cioabua2020spectral] used the variant of from [cioabaSpanningTrees] to show that a sufficient spectral condition for graph rigidity is essentially the best possible. We generalize their result in Section 5.

In Section 2, we will construct the family of graphs and prove the lower bound of Theorem 3. In Section 3, we will explicitly describe the characteristic polynomial of (Theorem 11), and in Section 4, we will use the characteristic polynomial to prove the upper eigenvalue bound of Theorem 3. The proof of the second eigenvalue bound uses a classical number theoretic technique, Graeffe’s method (see Lemma 12), which to the best of our knowledge has not previously been used for second eigenvalue bounds. This approach should generalize to upper bounds on the roots of interesting combinatorial polynomials.

## 2. Graph Construction

### 2.1. Construction

We construct a family of graphs such that , but for all . The graph contains copies of , each with a deleted matching of size . Then edges are added in a circulant manner to connect the vertices among the cliques with the deleted matchings.

Let . The vertex set of

consists of all ordered pairs

where and . Let , and let the subgraph induced by be , where

 Ei={(i,2a−2)∼(i,2a−1)∣1≤a≤m}.

Now we connect edges among the . Let

 E′={(i,2j+1)∼(i+j+1,2j)∣0≤i≤2m,0≤j≤m−1},

where is taken modulo . Then

 E(Gm,d)=2m⋃i=0E(G[Hi])∪E′.

This construction makes a connected -regular graph. See Figures 2.1 and 2.2.

### 2.2. Spanning Trees

Our result, like many prior results on edge-disjoint spanning trees, crucially relies on a theorem from Nash-Williams and Tutte, which converts a condition on to one on vertex partitions. If the vertex set is partitioned into disjoint sets , then let be the number of edges with endpoints in both and .

###### Theorem 4 (Nash-Williams/Tutte [nash1961edge, tutte1961problem]).

Let be a connected graph and be an integer. Then if and only if for any partition .

As in the previous subsection, let be the modified cliques of . Since , we have

 ∑0≤i

By Theorem 4, has at most edge-disjoint spanning trees. Then Theorem 2 implies , yielding the lower bound of Theorem 3. It remains to show .

## 3. Characteristic Polynomial

The adjacency matrix of is a block circulant matrix. Following [blockCirculantEvalues], define to be the block circulant matrix

where each is a square matrix of equal dimension.

###### Lemma 5 ([blockCirculantEvalues]).

The characteristic polynomial of a real, symmetric, block circulant matrix is given by

 det(xI−B(b0,b1,…,bn−1))=∏ζn=1det(xI−Hζ),

where

 Hζ=b0+ζb1+⋯+ζn−1bn−1,

and runs over the th roots of unity (including ).

To determine the characteristic polynomial of , we will also need the following lemmas from linear algebra. Let and

denote the identity matrix and all ones matrix of dimension

, respectively.

###### Lemma 6.

Let

be an invertible matrix and

be column vectors. Then,

 det(A+uvT)=(1+vTA−1u)det(A).
###### Lemma 7.

We have

 det(aIn+bJn) =an+nan−1b, (aIn+bJn)−1 =1aIn−ba(a+nb)Jn.

Finally, the characteristic polynomial of will require defining the Chebyshev polynomials of the first kind and Chebyshev polynomials of the second kind . We have [Abramowitz, p. 775, Equations (22.3.6) and (22.3.7)]

 (3.1) Tn(z) =n2⌊n/2⌋∑k=0(−1)kn−k(n−kk)(2z)n−2k, (3.2) Un(z) =⌊n/2⌋∑k=0(−1)k(n−kk)(2z)n−2k.

They are also given by the implicit equations and . We prove a few lemmas on the Chebyshev polynomials and their connection to roots of unity.

###### Lemma 8.

For ,

 T2m+1(z)z=22mm∏j=1(z2−sin2(πj2m+1)).
###### Proof.

Abramowitz and Stegun [Abramowitz, Page 787, Equation 22.16.4] provide the zeroes of as . The leading coefficient of is , as can be seen from equation (3.1), giving the product representation

 T2m+1(z) =22m2m+1∏ℓ=1(z−cos(ℓ−1/22m+1π)) =22mzm∏ℓ=1(z−cos(ℓ−1/22m+1π))(z+cos(ℓ−1/22m+1π)) =22mzm∏ℓ=1(z2−sin2(ℓ−m−12m+1π)) =22mzm∏ℓ=1(z2−sin2(ℓ2m+1π)).

###### Lemma 9.

For and any -th primitive root of unity ,

 m∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj) =T2m+1(z)z, m∑j=12x+ζj+¯¯¯¯¯ζjx2+2x−1+ζj+¯¯¯¯¯ζj =−12z+2m+12T2m+1(z)(T2m+1(z)−(z−1)U2m(z)),

with the change of variables .

###### Proof.

Applying Lemma 8,

 m∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj) =22mm∏j=1⎛⎜⎝(x+12)2−⎛⎝ζj/2−¯¯¯¯¯¯¯¯¯ζj/22ı⎞⎠2⎞⎟⎠ =22mm∏j=1((x+12)2−sin2(πj2m+1)) =T2m+1(x+12)(x+1)/2.

Also, we have

 m∑j=12x+ζj+¯¯¯¯¯ζjx2+2x−1+ζj+¯¯¯¯¯ζj =m∑j=1x+12−(ζj/2−¯¯¯¯¯¯¯¯ζj/22ı)2(x+12)2−(ζj/2−¯¯¯¯¯¯¯¯ζj/22ı)2 (3.3) =m∑j=1x+12−sin2(πj2m+1)(x+12)2−sin2(πj2m+1).

Taking a logarithm in Lemma 8 and then differentiating gives

 logT2m+1(z)−logz =log22m+m∑j=1log(z2−sin2(πj2m+1)), (3.4) (2m+1)U2m(z)T2m+1(z)−1z =2zm∑j=11z2−sin2(πj2m+1),

where we used for . This can be verified from the series expansions of both. Substituting equation (3.4) into equation (3.3) with gives

 m∑j=1z−sin2(πj2m+1)z2−sin2(πj2m+1) =m+(z−z2)m∑j=11z2−sin2(πj2m+1) =m+(1−z)(2m+1)2U2m(z)T2m+1(z)−1−z2z =−12z+2m+12T2m+1(z)(T2m+1(z)−(z−1)U2m(z)).

We generalize Lemma 9 for all roots of unity .

###### Lemma 10.

For and , let be a -th root of unity, where . Define and . Then

 m∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj) =(2Tn(z))g2z, m∑j=12x+ζj+¯¯¯¯¯ζjx2+2x−1+ζj+¯¯¯¯¯ζj =−12z+2m+12Tn(z)(Tn(z)−(z−1)Un−1(z)),

with the change of variables .

###### Proof.

Note that is a primitive th root of unity. By Lemma 9,

 n−12∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj) =Tn(z)z, n−12∑j=12x+ζj+¯¯¯¯¯ζjx2+2x−1+ζj+¯¯¯¯¯ζj =−12z+2m+12Tn(z)(Tn(z)−(z−1)Un−1(z)).

Notice . Using the periodicity of roots of unity several times,

 m∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj) =(n∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj))g−12⎛⎜ ⎜⎝n−12∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj)⎞⎟ ⎟⎠ =⎛⎜ ⎜⎝n−12∏j=1(x2+2x−1+ζj+¯ζj)⎞⎟ ⎟⎠g(x2+2x+1)g−12 =(Tn(z)z)g(x+1)g−1 =(2Tn(z))g2z.

Similarly,

 m∑j=12x+ζj+¯¯¯¯¯ζjx2+2x−1+ζj+¯¯¯¯¯ζj =gn−12∑j=12x+ζj+¯¯¯¯¯ζjx2+2x−1+ζj+¯¯¯¯¯ζj+g−122x+2x2+2x+1 =g(−12z+n2Tn(z)(Tn(z)−(z−1)Un−1(z)))+g−1x+1 =−12z+2m+12Tn(z)(Tn(z)−(z−1)Un−1(z)).

Finally, we compute the characteristic polynomial of . Denote to be the all ones matrix.

###### Theorem 11.

For , the characteristic polynomial of is

 (x+1)(d−2m)(2m+1)2m+1∏j=1(2Tn(z))g−1[(−2m+1+2m−dz)Tn(z)+(2m+1)(z−1)Un−1(z)],

with the change of variables , , and .

###### Proof.

By the symmetric construction of , its adjacency matrix is a block circulant matrix . For any , there is only one edge between and . Then for , is a matrix with a single entry of in position . Moreover, for . Finally, is the adjacency matrix of any . That is,

 b0=Jd+1−Id+1−2m∑i=1bi.

By Lemma 5, the eigenvalues of the adjacency matrix are the union of the eigenvalues of each

 Hζ =b0+ζb1+⋯+ζ2mb2m =[AJ2m,d−2m+1Jd−2m+1,2mJd−2m+1−Id−2m+1],

where is a -th root of unity and

 A=J2m+⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣−1¯¯¯ζ−1ζ−1−1−1¯¯¯¯¯ζ2−1ζ2−1−1⋱−1¯¯¯¯¯¯¯ζm−1ζm−1−1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

The characteristic polynomial is

 p(x)=∏ζ2m+1=1det(xI−Hζ).

We need the block determinant

Label the blocks . Passing to the Schur complement,

 (3.5) det(xI−Hζ)=det(D)det(A′−BD−1C).

By Lemma 7,

 (3.6) detD =(x+1)d−2m(x−d+2m), D−1 =1x+1I+1(x+1)(x−d+2m)J.

Note . Then,

 BD−1C =d−2m+1x+1J+(d−2m+1)2(x+1)(x−d+2m)J=d−2m+1x−d+2mJ, A′−BD−1C =−x+1x−d+2mJ+⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣x+11−¯¯¯ζ1−ζx+1x+11−¯¯¯¯¯ζ21−ζ2x+1⋱x+11−¯¯¯¯¯¯¯ζm1−ζmx+1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

Let be the block matrix summand of and be the all ones vector of dimension . Notice that , so that we can apply Lemma 6 to give

 (3.7) det(A′−BD−1C)=(1−x+1x−d+2muTM−1ζu)detMζ.

Here, gives the sum of the entries of . The inverse of a block diagonal matrix is the block diagonal matrix of the inverses of the blocks. Then the th block of is

 1x2+2x−1+ζj+¯¯¯¯¯ζj[x+1¯¯¯¯¯ζj−1ζj−1x+1],

and

 uTM−1ζu=m∑j=12x+ζj+¯¯¯¯¯ζjx2+2x−1+ζj+¯¯¯¯¯ζj.

The determinant of a block diagonal matrix is the product of the blocks’ determinants, so

 detMζ=m∏j=1(x2+2x−1+ζj+¯¯¯¯¯ζj).

By Lemma 10,

 uTM−1ζu =−12z+2m+12Tn(z)(Tn(z)−(z−1)Un−1(z)), detMζ =(2Tn(z))g2z.

Substituting into (3.7), we obtain

 det(A′−BD−1C)=(2Tn(z))g−1x−d+2m[(−2m+1+2m−dz)Tn(z)+(2m+1)(z−1)Un−1(z)].

With (3.6), we simplify (3.5) to

 det(xI−Hζ)=(x+1)d−2m(2Tn(z))g−1[(−2m+1+2m−dz)Tn(z)+(2m+1)(z−1)Un−1(z)].

Taking the product of over all roots of unity yields the characteristic polynomial. ∎

## 4. Bounding the second eigenvalue

Let denote the coefficient of in , and let be the roots of . Given any monic polynomial , we can rewrite as

 f(z)=n∑j=0ajzj=n−1∏j=0(z−zj)=zn−zn−1n−1∑j=0zj+O(zn−2).
###### Lemma 12 (Graeffe’s Method [Graeffe2]).

The largest root of a monic polynomial can be bounded by

 (4.1) z0≤(−4an−4+4an−3an−1+2a2n−2−4an−2a2n−1+a4n−1)1/4.
###### Proof.

We compute

 f(z)f(−z)=n−1∏j=0(z−zj)(z+zj)=n−1∏j=0(z2−z2j)=z2n−z2n−2n−1∑j=0z2j+O(z2n−4).

We know that this is an even function, so . Similarly, we can then compute

 f(z)f(−z)f(ız)f(−ız) =n−1∏j=0(z−zj)(z+zj)(z−ızj)(z+ızj)=n−1∏j=0(z4−z4j) =z4n−z4n−4n−1∑j=0z4j+O(z4n−8).

We conclude that

 z0≤(n−1∑j=0z4j)1/4=(−[z4n−4]f(z)f(−z)f(ız)f(−ız))1/4.

Moreover, only the powers of with degree at least multiply to produce a term of degree . By explicitly multiplying these polynomials out, we then have

 f(z)f(−z) f(ız)f(−ız) =4∏k=1((ıkz)n+an−1(ıkz)n−1+an−2(ıkz)n−2+an−3(ıkz)n−3+an−4(ıkz)n−4+O(zn−5)) =z4n−z4n−4(−4an−4+4an−3an−1+2a2n−2−4an−2a2n−1+a4n−1)+O(z4n−8).

The intuition behind why Graeffe’s method suffices here is that if the polynomial’s roots are well separated, then Graeffe’s method provides extremely precise approximations. Here, numerically we observed while the other roots were contained in . Then, passing to fourth powers using Graeffe’s method, the sum of fourth powers of all the roots is dominated by , so that taking a fourth root then provides a tight upper bound on the largest eigenvalue. We can use the same technique to derive a series of increasingly strong bounds on the largest root, as we incorporate more of the leading coefficients. For an introduction to the general method, see [Graeffe2]. For historical discussion of the origin of the name and method, see [Graeffe].

We require the following technical inequality.

###### Lemma 13.

For , we have

 (d4+4d3−(8m−2)d2+4d+8m2+4m+1)1/4
###### Proof.

Let and , so this statement reduces to showing

 ℓ4−4ℓ2n+2n2−6n+8ℓn<(ℓ−nℓ+2)4

for . By clearing denominators, expanding, and dividing through by , this is equivalent to showing

 2ℓ4(2n−5)+8ℓ3(n−6)−4ℓ2(n2+6n+12)−8ℓ(n2+8n−8)+n3−32n+96>0.

Note that for , so we drop this term completely and divide by . It is left to show that

 ℓ3(