The eigenvalues of Hessian matrices of the complete and complete bipartite graphs

TL;DR

This paper analyzes the eigenvalues of Hessian matrices for complete and complete bipartite graphs, revealing their Lorentzian metric properties and implications for algebraic structures.

Contribution

It computes eigenvalues of Hessian matrices for these graphs and demonstrates the Lorentzian nature and algebraic properties related to the strong Lefschetz property.

Findings

Eigenvalues include one positive and multiple negative values.

The Hessian matrices exhibit Lorentzian metric properties.

Strong Lefschetz property holds for related Artinian Gorenstein algebras.

Abstract

In this paper, we consider the Hessian matrices $H_{\Gamma}$ of the complete and complete bipartite graphs, and the special value of $\tilde H_{\Gamma}$ at $x_{i}=1$ for all $x_{i}$. We compute the eigenvalues of $\tilde H_{\Gamma}$. We show that one of them is positive and that the others are negative. In other words, the metric with respect to the symmetric matrix $\tilde H_{\Gamma}$ is Lorentzian. Hence those Hessian $\det (H_{\Gamma})$ are not identically zero. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the graphic matroids of the complete and complete bipartite graphs with at most five vertices.

Equations

$$\times L^{r-2p}:B^p(M)\to B^{r-p}(M)$$

$$F_{\Gamma }=\sum _{T\in B_{\Gamma }}\prod _{e\in E(T)}{x}_e.$$

$$H_{\Gamma }={\left(\frac{\partial }{\partial x_i}\frac{\partial }{\partial x_j}{F}_{\Gamma }\right)}_{i,j\in E(\Gamma )}$$

$$A={\left(A^{ij}\right)}_{1\le i,j\le l}$$

$$\bar{A}{\left(w_i\right)}_{1\le i\le l}=\lambda {\left(w_i\right)}_{1\le i\le l}.$$

$$\left(\begin{array}{cccc}{\bar{a}}_{i1}& {\bar{a}}_{i2}& \cdots & {\bar{a}}_{il}\end{array}\right)\left(\begin{array}{c}{w}_1\\ w_2\\ ⋮\\ w_l\end{array}\right)$$

$$w_1{A}^{i1}{\mathbf{v}}_1+w_2{A}^{i2}{\mathbf{v}}_2+\cdots +w_l{A}^{il}{\mathbf{v}}_l$$

$$=({\bar{a}}_{i1}{w}_1+{\bar{a}}_{i2}{w}_2+\cdots +{\bar{a}}_{il}{w}_l){\mathbf{v}}_i$$

$$=\left(\left(\begin{array}{cccc}{\bar{a}}_{i1}& {\bar{a}}_{i2}& \cdots & {\bar{a}}_{il}\end{array}\right)\left(\begin{array}{c}{w}_1\\ w_2\\ ⋮\\ w_l\end{array}\right)\right){\mathbf{v}}_i$$

$$=\lambda w_i{\mathbf{v}}_i$$

$$A\mathbf{x}$$

$$=\left(\begin{array}{c}{w}_1{A}^{11}{\mathbf{v}}_1+w_2{A}^{12}{\mathbf{v}}_2+\cdots +w_l{A}^{1l}{\mathbf{v}}_l\\ ⋮\\ w_1{A}^{l1}{\mathbf{v}}_1+w_2{A}^{l2}{\mathbf{v}}_2+\cdots +w_l{A}^{ll}{\mathbf{v}}_l\end{array}\right)$$

$$=\lambda \left(\begin{array}{c}{w}_1{\mathbf{v}}_1\\ w_2{\mathbf{v}}_2\\ ⋮\\ w_l{\mathbf{v}}_l\end{array}\right)$$

$$=\lambda \mathbf{x}.$$

$$\left(\begin{array}{ccccc}0& 1& & & \\ & 0& 1& & \\ & & \ddots & \ddots & \\ & & & 0& 1\\ 1& & & & 0\end{array}\right)$$

$${\mathbf{z}}_{n,k}=\left(\begin{array}{c}1\\ {\zeta }_n^k\\ {\zeta }_n^{2k}\\ ⋮\\ {\zeta }_n^{(n-1)k}\end{array}\right).$$

$${\chi }_A(t)$$

$$\det A$$

$${\chi }_A(t)$$

$$\det A$$

$${\bar{C}}_k={\left(c_{ij}^{(k)}\right)}_{1\le i,j\le l}.$$

$${\chi }_C(t)$$

$$\det C$$

$$D^{il}=\left(\begin{array}{c}{X}_i\\ X_i\end{array}\right),$$

$$d_{ij}^{(k)}=\{\begin{array}{ll}(D^{ij}{)}^{(1)}{\mathbf{z}}_{2n,k}& \text{if 1≤j≤l−1},\\ (D^{ij}{)}^{(1)}{\mathbf{z}}_{n,\frac{k}{2}}& \text{if j=l}.\end{array}$$

$$d_{ij}^{(k)}=\{\begin{array}{ll}(D^{ij}{)}^{(1)}{\mathbf{z}}_{2n,k}& \text{if 1≤j≤l−1},\\ 0& \text{if j=l}.\end{array}$$

$$\left(\begin{array}{c}{w}_1{\mathbf{z}}_{2n,k}\\ ⋮\\ w_{l-1}{\mathbf{z}}_{2n,k}\\ w_l{\mathbf{z}}_{n,\frac{k}{2}}\end{array}\right)\in {\mathbb{C}}^{2(l-1)n}$$

$${\mathbf{z}}_{2n,k}=\left(\begin{array}{c}{\mathbf{z}}_{n,\frac{k}{2}}\\ {\mathbf{z}}_{n,\frac{k}{2}}\end{array}\right).$$

$$\left(\begin{array}{c}{0}_{l-1}\\ 1\end{array}\right),$$

$$\left(\begin{array}{c}{w}_1{\mathbf{z}}_{2n,k}\\ ⋮\\ w_{l-1}{\mathbf{z}}_{2n,k}\\ 0_n\end{array}\right)\in {\mathbb{C}}^{2(l-1)n}$$

$$\displaystyle\left(\begin{array}[]{c}\boldsymbol{0}_{l-1}\\ 1\end{array}\right)$$

$${\chi }_D(t)$$

$$=\frac{1}{t^n}\prod _{k=0}^{2n-1}{\chi }_{{\bar{D}}_k}(t).$$

$$T(A,\mathbf{d})$$

$$D(\lambda ,\mathbf{d})$$

$$M(A,\lambda ,\mathbf{d})=T(A,\mathbf{d})+D(\lambda ,\mathbf{d}).$$

Full text

These research results were aided by the fund of Nagano Prefecture to promote scientific activity.

1. Introduction

In [5], Maeno and Numata introduced algebras $Q/J_{M}$ and $A_{M}$ for a matroid $M$ to give an algebraic proof of the Sperner property for the lattice $\mathcal{L}(M)$ consisting of flats of $M$. The algebra $A_{M}$ is defined to be the quotient algebra of the ring of the differential polynomials by the annihilator of the base generating function of $M$. By definition, the algebra $A_{M}$ is an Artinian Gorenstein algebra. In [5], they show that $Q/J_{M}$ has the strong Lefshetz property in the narrow sense (see Definition 4.1 for the definition, see also [2]) if and only if the lattice $\mathcal{L}(M)$ is modular, and that $Q/J_{M}=A_{M}$ if and only if $\mathcal{L}(M)$ is modular. In [3], for a simple matroid $M$ with rank $r$ and the ground set $E$, Huh and Wang introduced a graded algebra $B^{\ast}(M)=\bigoplus_{p=0}^{r}B^{p}(M)$, which is isomorphic to $Q/J_{M}$. They show that the multiplication map

[TABLE]

is injective for $p\leq\frac{r}{2}$ and $L=\sum_{i\in E}y_{i}$.

Maeno and Numata conjectured that the algebra $A_{M}$ has the strong Lefschetz property for an arbitrary matroid $M$ in an extended abstract [4] of the paper [5]. In this paper, we discuss the strong Lefschetz property of the Artinian Gorenstein algebra $A_{M}$ for graphic matroids of the complete and the complete bipartite graphs.

Let us recall the definition of $A_{M}$ in the case of a graphic matroids. Let $\Gamma$ be a connected graph with $N$ edges. Define $B_{\Gamma}$ to be the set of spanning trees in $\Gamma$. Then the graphic matroid $M(\Gamma)$ of $\Gamma$ is a matroid whose ground set is the set $E(\Gamma)$ of edges $\Gamma$ and whose basis set is the set $B_{\Gamma}$ of spanning trees in $\Gamma$. We assign the variable $x_{i}$ to each edge $i$ of $\Gamma$. For the graph $\Gamma$, we define the weighted generating function $F_{\Gamma}$ for spanning trees in $\Gamma$, called Kirchhoff polynomial of $\Gamma$, by

[TABLE]

Then $A_{M(\Gamma)}$ is $\mathbb{K}[x_{1},\ldots,x_{N}]/\operatorname{Ann}(F_{\Gamma})$ (see Section 4 for the definition of $\operatorname{Ann}(F_{\Gamma})$).

There is a criterion for the strong Lefschetz property of a graded Artinian Gorenstein algebra (see Theorem 4.3 for the detail). Roughly speaking, a graded Artinian Gorenstein algebra $A=\bigoplus_{k=0}^{s}A_{k}$ has the strong Lefshetz property if and only if the determinants of the $k$th Hessian matrices of the algebra do not vanish for all $k$, where the $k$th Hessian matrix is obtained from a linear basis for the homogeneous space $A_{k}$ of degree $k$.

Now we define the matrix $H_{\Gamma}$ by

[TABLE]

for a connected graph $\Gamma$. We call $H_{\Gamma}$ the Hessian matrix of the graph $\Gamma$ and $\det(H_{\Gamma})$ the Hessian of the graph $\Gamma$. If the determinant of $H_{\Gamma}$ does not vanish, then the set of all variables is a linear basis for the homogeneous space of degree one. Hence the Hessian matrix of the graph coincidences with the first Hessian matrix of the algebra $A_{M(\Gamma)}$.

In this paper, we consider the Hessian matrices $H_{\Gamma}$ of the complete and complete bipartite graphs, and the special value of $\tilde{H}_{\Gamma}$ at $x_{i}=1$ for all $x_{i}$. We compute the eigenvalues of $\tilde{H}_{\Gamma}$. We show that one of them is positive and that the others are negative. In other words, the metric with respect to the symmetric matrix $\tilde{H}_{\Gamma}$ is Lorentzian. Hence those Hessian $\det(H_{\Gamma})$ are not identically zero. This implies the strong Lefschetz property of the Artinian Gorenstein algebra corresponding to the graphic matroid of the complete graph and the complete bipartite graph with at most five vertices.

This article is organized as follows: In Section 2, we will calculate the eigenvectors and eigenvalues of some block matrices. Then we will compute the Hessians of the complete graph and the complete bipartite graph in Section 3. In Section 4, we will discuss the strong Lefschetz property of the algebra $A_{M}$ corresponding to a graphic matroid.

2. The eigenvectors and eigenvalues of block matrices

In this section, we give the eigenvectors and eigenvalues of some block matrices. We consider three kinds of block matrices $C,D$ and $M(A,\lambda,\boldsymbol{d})$ (Theorems 2.5, 2.9 and 2.11).

Let $l\in\mathbb{Z}$, $\boldsymbol{d}=(d_{1},d_{2},\ldots,d_{l})$, and $\delta=d_{1}+d_{2}+\cdots+d_{l}$. Let $A^{ij}$ be a $d_{i}\times d_{j}$ matrix. We consider the $\delta\times\delta$ matrix $A$ defined by

[TABLE]

with the $d_{i}\times d_{j}$ matrix $A^{ij}$.

Lemma 2.1**.**

Let $A^{ij}\boldsymbol{v}_{j}=\bar{a}_{ij}\boldsymbol{v}_{i}$ and $\bar{A}=(\bar{a}_{ij})_{1\leq i,j\leq l}$, where $\boldsymbol{v}_{i}$ and $\boldsymbol{v}_{j}$ are vectors of length $d_{i}$ and $d_{j}$, respectively. If $\left(w_{i}\right)_{1\leq i\leq l}\in\mathbb{C}^{l}$ is an eigenvector of $\bar{A}$ belonging to the eigenvalue $\lambda$, then $\boldsymbol{x}=\left(w_{i}\boldsymbol{v}_{i}\right)_{1\leq i\leq l}\in\mathbb{C}^{\delta}$ satisfies $A\boldsymbol{x}=\lambda\boldsymbol{x}$.

Proof.

Since $\left(w_{i}\right)_{1\leq i\leq l}$ is the eigenvector of $\bar{A}$ belonging to $\lambda$,

[TABLE]

The $i$th row of the equation implies that

[TABLE]

for all $i$. Since $A^{ij}\boldsymbol{v}_{j}=\bar{a}_{ij}\boldsymbol{v}_{i}$,

[TABLE]

for all $i$. Hence

[TABLE]

∎

Remark 2.2**.**

The vector $\boldsymbol{x}$ may be equal to the zero vector $\boldsymbol{0}_{\delta}$ of size $\delta$. Hence the vector $\boldsymbol{x}$ may not be an eigenvector of $A$.

Let $C_{n}$ be the square matrix

[TABLE]

of size $n$. Let $\zeta_{n}$ be the $n$th primitive root of unity, and

[TABLE]

Then $\boldsymbol{z}_{n,k}$ is an eigenvector of $C_{n}$ belonging to the eigenvalue $\zeta_{n}^{k}$. Note that $\boldsymbol{z}_{n,k}$ is also an eigenvector of $(C_{n})^{t}$ belonging to the eigenvalue $\zeta_{n}^{kt}$.

Let $I_{n}$ be the identity matrix of size $n$, and $J_{mn}$ the all-one matrix of size $m\times n$. For an $n\times n$ matrix $X$, $\chi_{X}(t)$ denotes the characteristic polynomial $\det(tI_{n}-X)$ of $X$ in the variable $t$, and $X^{(1)}$ denotes the first row of $X$. Note that the product $X^{(1)}\boldsymbol{z}_{n,k}$ is a scalar.

Lemma 2.3**.**

Let $A$ be an $n\times n$ cyclic matrix. The vector $\boldsymbol{z}_{n,k}$ is an eigenvector of $A$ belonging to the eigenvalue $A^{(1)}\boldsymbol{z}_{n,k}$. Hence

[TABLE]

We obtain the following from Lemma 2.3

Lemma 2.4**.**

Let $\alpha,\lambda\in\mathbb{C}$ and $A=\alpha J_{nn}+\lambda I_{n}$. The vector $\boldsymbol{z}_{n,0}$ is an eigenvector of $A$ belonging to the eigenvalue $\lambda+n\alpha$. For $1\leq k\leq n-1$, the vector $\boldsymbol{z}_{n,k}$ is an eigenvector of $A$ belonging to the eigenvalue $\lambda$. Hence

[TABLE]

Now we consider the block matrices $C,D$ and $M(A,\lambda,\boldsymbol{d})$.

First we consider the case where $\boldsymbol{d}=(n,\ldots,n)$ and each block is cyclic. Let us consider the block matrix $C$ whose blocks are $n\times n$ cyclic matrices. Let $C$ be $\left(C^{ij}\right)_{1\leq i,j\leq l}$, $C^{ij}$ an $n\times n$ matrix for each $i,j$, and $c_{ij}^{(k)}$ an eigenvalue of $C^{ij}$ associated with an eigenvector $\boldsymbol{z}_{n,k}$. For $C$ and $0\leq k\leq n-1$, we define the $l\times l$ matrix $\bar{C}_{k}$ by

[TABLE]

Theorem 2.5**.**

Let $\left(w_{i}\right)_{1\leq i\leq l}\in\mathbb{C}^{l}$ be an eigenvector of $\bar{C}_{k}$ belonging to the eigenvalue $\lambda$. Then $\left(w_{i}\boldsymbol{z}_{n,k}\right)_{1\leq i\leq l}\in\mathbb{C}^{nl}$ is an eigenvector of $C$ belonging to $\lambda$. Hence

[TABLE]

Proof.

Since $\boldsymbol{z}_{n,k}$ is a nonzero vector for any $k$, the vector $\left(w_{i}\boldsymbol{z}_{n,k}\right)_{1\leq i\leq l}$ is also a nonzero vector. Since $C^{ij}$ is cyclic, it follows from Lemma 2.3 that $\bar{C}_{k}$ satisfies the assumption in Lemma 2.1. ∎

Next we consider the case where $\boldsymbol{d}=(2n,2n,\ldots,2n,n)$. Let $D$ be the block matrix $D=\left(D^{ij}\right)_{1\leq i,j\leq l}$. We assume that $D^{ij}$ is a $2n\times 2n$ cyclic matrix if $1\leq i,j\leq l-1$ and that $D^{ll}$ is an $n\times n$ cyclic matrix. Moreover we assume that

[TABLE]

for $1\leq i,j\leq l-1$, where $X_{i}$ and $Y_{j}$ are $n\times n$ cyclic matrices.

For $0\leq k\leq 2n-1$, we define the $l\times l$ matrix $\bar{D}_{k}=(d_{ij}^{(k)})_{1\leq i,j\leq l}$ as follows: If $k$ is even, then we define

[TABLE]

If $k$ is odd, then

[TABLE]

Lemma 2.6**.**

Let $0\leq k\leq 2n-1$ and $k$ be even. Let $\left(w_{i}\right)_{1\leq i\leq l}\in\mathbb{C}^{l}$ be an eigenvector of $\bar{D}_{k}$ belonging to the eigenvalue $\lambda$. Then

[TABLE]

is an eigenvector of $D$ belonging to $\lambda$.

Proof.

Now the matrix $D$ satisfies the equation (1), and we have

[TABLE]

Hence Lemmas 2.1 and 2.3 imply Lemma 2.6. ∎

Lemma 2.7**.**

Let $0\leq k\leq 2n-1$ and $k$ be odd. Let $\boldsymbol{w}=\left(w_{i}\right)_{1\leq i\leq l}$ be an eigenvector of $\bar{D}_{k}$ belonging to the eigenvector $\lambda$. If the vector $\boldsymbol{w}$ is linearly independent of the vector

[TABLE]

then

[TABLE]

is an eigenvector of $D$ belonging to $\lambda$.

Proof.

It can be also shown by Lemmas 2.1 and 2.3. ∎

Remark 2.8**.**

The vector

[TABLE]

is an eigenvector of $\bar{D}_{k}$ belonging to the eigenvalue zero. The eigenvector, however, does not induce an eigenvector of $D$. See also Remark 2.2.

Theorem 2.9 follows from Lemmas 2.6 and 2.7

Theorem 2.9**.**

The characteristic polynomial of $D$ is

[TABLE]

Finally we consider the block matrix $M(A,\lambda,\boldsymbol{d})$. For a square matrix $A$ of size $l$, $\boldsymbol{d}=(d_{1},d_{2},\ldots,d_{l})$, and $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{l})$, we define

[TABLE]

We define the square matrix $M(A,\lambda,\boldsymbol{d})$ of size $d_{1}+\cdots+d_{l}$ by

[TABLE]

We define also the square matrix $\bar{M}(A,\lambda,\boldsymbol{d})$ of size $l$ by

[TABLE]

where $\operatorname{diag}(x_{1},\ldots,x_{l})$ is the diagonal matrix with entries $x_{1},\ldots,x_{l}$.

Lemma 2.10**.**

Let $\lambda=(\lambda_{1},\ldots,\lambda_{l})$, $\boldsymbol{d}=(d_{1},d_{2},\ldots,d_{l})$, and $\delta=d_{1}+d_{2}+\cdots+d_{l}$.

1. (1)

Let $i\in\Set{1,2,\ldots,l}$. Let $1\leq k\leq d_{i}-1$. We suppose that $w_{i}=1$ and that for $j\in\Set{1,2,\ldots,l}\setminus\Set{i},w_{j}=0$. The vector $\left(w_{j}\boldsymbol{z}_{d_{j},k}\right)_{1\leq j\leq l}\in\mathbb{C}^{\delta}$ is an eigenvector of $M(A,\lambda,\boldsymbol{d})$ belonging to the eigenvalue $\lambda_{i}$. 2. (2)

If $\left(w_{i}\right)_{1\leq i\leq l}\in\mathbb{C}^{l}$ is an eigenvector of $\bar{M}(A,\lambda,\boldsymbol{d})$ belonging to the eigenvalue $\mu$, then $\left(w_{i}\boldsymbol{z}_{d_{j},0}\right)_{1\leq i\leq l}\in\mathbb{C}^{\delta}$ is an eigenvector of $M(A,\lambda,\boldsymbol{d})$ belonging to the eigenvalue $\mu$.

Proof.

Let $A=(a_{ij})$. Since $1\leq k\leq d_{i}-1$, it holds that $\zeta_{d_{i}}^{k}\neq 1$. Hence

[TABLE]

Therefore the claim $(1)$ follows from Lemmas 2.1 and 2.4. It follows that

[TABLE]

for $i\neq j$, and that

[TABLE]

for all $i$. Then the claim $(2)$ follows from Lemma 2.1. ∎

Theorem 2.11 follows from Lemma 2.10.

Theorem 2.11**.**

For a matrix $A$ of size $l$, $\lambda=(\lambda_{1},\ldots,\lambda_{l})$ and $\boldsymbol{d}=(d_{1},d_{2},\ldots,d_{l})$, we have

[TABLE]

3. Main results

In this section, we will compute the Hessian matrices of the complete graph and the complete bipartite graph, defined in Section 1. We define $\tilde{H}_{\Gamma}$ to be the special value of $H_{\Gamma}$ at $x_{i}=1$ for all $i$. Note that $(i,j)$-entry in $\tilde{H}_{\Gamma}$ is the number of the spanning trees including edges $i,j$. First we compute the number of spanning trees containing the edges. Next we compute $\det\tilde{H}_{\Gamma}$ by theorems in Section 2.

Terms of graphs in this section follows mainly [1].

3.1. The Hessian of the complete graph

Here we compute the Hessians of the complete graphs (Theorem 3.4).

Let $n\geq 3$. The $(i,j)$-entry in $\tilde{H}_{K_{n}}$ is the number of trees including edges $i,j$ in $K_{n}$ with $n$ vertices. Moon gave the following formula for the number of trees containing a prescribed set of edges.

Theorem 3.1** (Moon [7]).**

Let $F$ be a forest with $k$ connected components. The number of the trees with $n$ vertices containing $F$ is

[TABLE]

where $j_{i}$ is the number of vertices of each component of $F$.

Let $\tilde{H}_{K_{n}}=(h_{ij})$. It follows from Theorem 3.1 that

[TABLE]

We can prove the following.

Theorem 3.2**.**

*The eigenvalues of $\tilde{H}_{K_{n}}$ are $-2n^{n-4}$, $-n^{n-3}$ and $2(n-2)n^{n-3}$. The dimensions of the eigenspaces associate with $-2n^{n-4}$, $-n^{n-3}$ and $2(n-2)n^{n-3}$ are $\binom{n}{2}-n$, $n-1$ and $1$, respectively. *

Remark 3.3**.**

Theorem 3.2 implies that one of the eigenvalues of $\tilde{H}_{K_{n}}$ is positive and that the others are negative.

Theorem 3.2 implies Theorem 3.4.

Theorem 3.4**.**

Let $n\geq 3$. Then the determinant of $\tilde{H}_{K_{n}}$ is

[TABLE]

Hence the Hessian $\det H_{K_{n}}$ does not vanish for $n\geq 3$.

Now we prove Theorem 3.2. Let $n\geq 3$. We define $a(e,e^{\prime})$ by

[TABLE]

for $e,e^{\prime}\in E(K_{n})$.

Let $V(K_{n})=\Set{0,1,\ldots,n-1}$. We consider a group action on $V(K_{n})$ as follows: Let $G$ be the cyclic group generated by $\sigma$ of the order $n$. For a vertex of $K_{n}$, define

[TABLE]

The action of $G$ on $V(K_{n})$ induces an action on $E(K_{n})$ by

[TABLE]

for $\Set{i,j}\in E(K_{n})$.

Under this notation, first we show Theorem 3.2 in the case where $n$ is odd. And then, we show Theorem 3.2 in the case where $n$ is even.

Consider the case where $n$ is odd. Let $n=2l+1$. For $1\leq i,j\leq l$, we define

[TABLE]

where $e_{i}=\Set{0,i}\in E(K_{n})$. Note that $C^{ij}$ is an $n\times n$ cyclic matrix by the definition of the action. Since

[TABLE]

the matrix $n^{n-4}C$ is $\tilde{H}_{K_{n}}$. Let us calculate eigenvalues of $C$ by Theorem 2.5. For $0\leq k\leq n-1$, let

[TABLE]

First we consider the case where $1\leq k\leq n-1$.

Lemma 3.5**.**

Let $1\leq k\leq n-1$. Then

[TABLE]

where

[TABLE]

for all $i,j$. Moreover the rank of $\bar{C}_{k}-(-2)I_{l}$ is one.

Proof.

We fix $k$ and compute $(C^{ij})^{(1)}\boldsymbol{z}_{n,k}$. First we consider the case where $e_{i}\neq e_{j}$. The edges $e_{i}$ and $\sigma^{l}e_{j}$ share their vertices if and only if $l=0$, $l=i$, $j+l=0$, and $j+l=i$. Since $e_{i}\neq e_{j}$, there does not exist $l$ such that $e_{i}=\sigma^{l}e_{j}$. Hence if $l\in\Set{0,i,-j,i-j}$, then

[TABLE]

and if $l\notin\Set{0,i,-j,i-j}$, then

[TABLE]

Therefore

[TABLE]

Next we consider the case where $e_{i}=e_{j}$. The edges $e_{i}$ and $\sigma^{l}e_{i}$ share their vertices if and only if $l=0,l=i$ and $l+i=0$. If $l=0$, then

[TABLE]

if $l=i$ or $l=-i$, then

[TABLE]

Hence if $l\notin\Set{0,i,-i}$, then

[TABLE]

Therefore

[TABLE]

We have

[TABLE]

Hence the rank of $\bar{C}_{k}-(-2)I_{l}$ is one.

∎

Next we obtain all eigenvalues of $\bar{C}_{k}$ for $1\leq k\leq n-1$ by computing the trace of $\bar{C}_{k}$.

Lemma 3.6**.**

For $1\leq k\leq n-1$, the eigenvalues of $\bar{C}_{k}$ are $-2$ and $-n$. The dimensions of the eigenspaces of $\bar{C}_{k}$ associate with $-2$ and $-n$ are $l-1$ and $1$, respectively.

Proof.

The trace of $\bar{C}_{k}$ is

[TABLE]

Therefore it follows from Lemmas 2.5 and 3.5 that the eigenvalues of $\bar{C}_{k}$ are $-2$ and $-n$, and the dimensions of the eigenspaces of $\bar{C}_{k}$ associate with $-2$ and $-n$ are $l-1$ and $1$, respectively. ∎

Similarly, we obtain the consequence in the case where $k=0$.

Lemma 3.7**.**

The eigenvalues of $\bar{C}_{0}$ are $-2$ and $2n(n-2)$. The dimensions of the eigenspaces of $\bar{C}_{0}$ associate with $-2$ and $2n(n-2)$ are $l-1$ and $1$, respectively.

We obtain the following result by Theorem 2.5, and Lemmas 3.6 and 3.7.

Lemma 3.8**.**

The eigenvalues of $C$ are $-2$, $-n$ and $2n(n-2)$. The dimensions of the eigenspaces of $C$ associate with $-2$, $-n$ and $2n(n-2)$ are $\binom{n}{2}-n$, $n-1$ and $1$, respectively.

Next we consider the case where $n$ is even. Let $n=2l$. We define the matrix $D^{ij}$ by

[TABLE]

and the block matrix $D$ by $D=\left(D^{ij}\right)_{1\leq i,j\leq l}$. Note that the matrix $n^{n-4}D$ is $\tilde{H}_{K_{n}}$ since

[TABLE]

For $0\leq j\leq l-1$, the $(1,k)$ entry and $(1,k+l)$ entry in the $(l,j)$ block satisfy

[TABLE]

For $0\leq i\leq l-1$, the $(i,l)$ block also satisfies the assumption (1) since $D$ is a symmetric matrix. Hence the matrix $D$ in this section satisfies the assumption of the matrix $D$ in Section 2.

Let us calculate eigenvalues of $D$ by Lemmas 2.6 and 2.7. For $0\leq k\leq 2n-1$, we define the $l\times l$ matrix $\bar{D}_{k}=(d_{ij}^{(k)})_{1\leq i,j\leq l}$ as follows: If $k$ is even, then we define

[TABLE]

If $k$ is odd, then we define

[TABLE]

First we consider the case where $1\leq k\leq 2n-1$.

Lemma 3.9**.**

Let $1\leq k\leq 2n-1$. Then

[TABLE]

where

[TABLE]

for all $i$, and

[TABLE]

Moreover the rank of $\bar{D}_{k}-(-2)I_{l}$ is one.

Proof.

Let $k$ be even. We compute the entries $(D^{ij})^{(1)}\boldsymbol{z}_{n,k}$, $(D^{il})^{(1)}\boldsymbol{z}_{l,\frac{k}{2}}$ and $(D^{ll})^{(1)}\boldsymbol{z}_{l,\frac{k}{2}}$ in $\bar{D}_{k}$.

First we consider the case where $1\leq j\leq l-1$. In this case we can show the Lemma in the same way as Lemma 3.5.

Next let us consider the case where $1\leq i\leq l-1$ and $j=l$. We compute $(D^{il})^{(1)}\boldsymbol{z}_{l,\frac{k}{2}}$ for $1\leq i\leq l-1$. In this case $j=-j$. The edges $e_{i}$ and $\sigma^{s}e_{l}$ share their vertices if and only if $s=0$, $s=i$. Since $e_{i}\neq e_{l}$, there does not exist $s$ such that $e_{i}=\sigma^{s}e_{l}$. Hence if $s\in\Set{0,i}$, then

[TABLE]

and if $s\notin\Set{0,i}$, then

[TABLE]

Therefore

[TABLE]

Finally we consider the case where $i=j=l$. If $s=0$, then

[TABLE]

If $s\neq 0$, then $e_{l}$ and $\sigma^{s}e_{l}$ do not share their vertices. Hence if $s\neq 0$, then

[TABLE]

Therefore

[TABLE]

Let $k$ be odd. The $l$th column of $\bar{D}_{k}$ is the zero vector by definition. On the other hand, since $k$ is odd, $\zeta_{n}^{lk}=-1$. Hence $\xi^{\prime}_{l}=0$.

Similarly to Lemma 3.5, we can show that the rank of $\bar{D}_{k}-(-2)I_{l}$ is one. ∎

Lemma 3.10**.**

Let $1\leq k\leq 2n-1$ and $k$ be even. The eigenvalues of $\bar{D}_{k}$ are $-2$ and $-n$. The dimensions of the eigenspaces of $\bar{D}_{k}$ associate with $-2$ and $-n$ are $l-1$ and $1$, respectively.

Proof.

The trace of $\bar{D}_{k}$ is

[TABLE]

Therefore it follows from Lemmas 2.6 and 3.9 that the eigenvalues of $\bar{D}_{k}$ are $-2$ and $-n$, and the dimensions of the eigenspaces of $\bar{D}_{k}$ associate with $-2$ and $-n$ are $l-1$ and $1$, respectively. ∎

Lemma 3.11**.**

Let $1\leq k\leq 2n-1$ and $k$ be odd. The eigenvalues of $\bar{D}_{k}$ are $-2$, $-n$ and [math]. The dimensions of the eigenspaces of $\bar{D}_{k}$ associate with $-2$, $-n$ and [math] are $l-2$, $1$ and $1$, respectively.

Proof.

Similarly to Lemma 3.10, we compute the trace of $\bar{D}_{k}$ and apply Lemma 2.7 and Remark 2.8. ∎

Similarly, we obtain the consequence in the case where $k=0$.

Lemma 3.12**.**

The eigenvalues of $\bar{D}_{0}$ are $-2$ and $2n(n-2)$. The dimensions of the eigenspaces of $\bar{D}_{0}$ associate with $-2$ and $2n(n-2)$ are $l-1$ and $1$, respectively.

We obtain the following result by Theorem 2.9, and Lemmas 3.10, 3.11 and 3.12.

Lemma 3.13**.**

The eigenvalues of $D$ are $-2$, $-n$ and $2n(n-2)$. The dimensions of the eigenspaces of $D$ associate with $-2$, $-n$ and $2n(n-2)$ are $\binom{n}{2}-n$, $n-1$ and $1$, respectively.

On combining Lemma 3.8 with Lemma 3.13, we obtain Theorem 3.2.

3.2. The Hessian of the complete bipartite graph

Here we compute the Hessians of the complete bipartite graphs (Theorem 3.19).

For a graph $\Gamma$, we define the degree matrix $D_{\Gamma}$ to be a diagonal matrix indexed by vertices of $\Gamma$ whose entries are degrees of vertices. We also define the adjacency matrix $A_{\Gamma}$ to be a matrix indexed by vertices of $\Gamma$ whose $(i,j)$-entry is the number of edges with the ends $v_{i}$ and $v_{j}$. Note that the entries in $A_{\Gamma}$ are one or zero for a simple graph $\Gamma$. We define the Laplacian matrix $L_{\Gamma}$ by $L_{\Gamma}=D_{\Gamma}-A_{\Gamma}$. For an arbitrary graph, we have the theorem which is the number of spanning trees in the graph. See [1, Theorem 6.3] for the detail.

Theorem 3.14** (The matrix-tree theorem).**

Every cofactor of $L_{\Gamma}$ is equal to the number of spanning trees in $\Gamma$.

For a graph $\Gamma$ and an edge $e$ with ends $v,v^{\prime}$ of $\Gamma$, we define the contraction $\Gamma/e$ to be the graph obtained by removing the edge $e$ from $\Gamma$ and by putting $v$ in $v^{\prime}$. Let $e,e^{\prime}$ be edges of $\Gamma$. Then $(\Gamma/e)/e^{\prime}=(\Gamma/e^{\prime})/e$. We write $\Gamma/e,e^{\prime}$ to denote $(\Gamma/e)/e^{\prime}$.

Let $e$ be an edge of a graph $\Gamma$. If we apply the matrix-tree theorem to the graph $\Gamma/e$, then we obtain the number of spanning trees in $\Gamma$ including the edge $e$.

Let $X=\Set{0^{\prime},1^{\prime},\ldots,(m-1)^{\prime}}$ and $Y=\Set{0,1,\ldots,n-1}$. We consider the complete bipartite graph $K_{m,n}=K_{X,Y}$. Theorems 2.11 and 3.14 implies the following.

Lemma 3.15**.**

Let $h_{ij}$ be the number of spanning trees containing the edges $i,j,i\neq j$ of $K_{m,n}$. Then

[TABLE]

Proof.

The Laplacian matrix $L_{K_{m,n}}$ is

[TABLE]

where $M$ is as in Section 2. The number of spanning trees including edges $i,j$ is every cofactor of the graph obtained by contraction edges $i,j$ of $K_{m,n}$. If $i\cap j\in X$, then the Laplacian matrix $L_{K_{m,n}/i,j}$ is

[TABLE]

where

[TABLE]

It follows from Proposition 2.11 that the $(1,1)$ cofactor of $L_{K_{m,n}/i,j}$ is

[TABLE]

Similarly if $i\cap j\in Y$, then the $(1,1)$ cofactor of $L_{K_{m,n}/i,j}$ is

[TABLE]

Finally we consider the case of $i\cap j=\emptyset$. Then the Laplacian matrix $L_{K_{m,n}/i,j}$ is

[TABLE]

where

[TABLE]

By Proposition 2.11, the $(1,1)$ cofactor of $L_{K_{m,n}/i,j}$ is

[TABLE]

∎

Example 3.16**.**

Consider the complete bipartite graph $K_{2,3}$, $(m=2,n=3)$. The following are the subgraphs of $K_{2,3}$ consisting four edges including the edges $\Set{0^{\prime},0}$ and $\Set{0^{\prime},1}$:

\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]

Hence the number of spanning tree containing the edges $\Set{0^{\prime},0}$ and $\Set{0^{\prime},1}$ are five.

The following are the subgraphs of $K_{2,3}$ consisting four edges including the edges $\Set{0^{\prime},0}$ and $\Set{1^{\prime},0}$:

\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]

Hence the number of spanning tree containing the edges $\Set{0^{\prime},0}$ and $\Set{1^{\prime},0}$ are four.

The following are the subgraphs of $K_{2,3}$ consisting four edges including the edges $\Set{0^{\prime},0}$ and $\Set{1^{\prime},1}$:

\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]\bullet$$\bullet$$\circ$$\circ$$\circ$$1^{\prime}$$0^{\prime}$$2$$1[math]

Hence the number of spanning tree containing the edges $\Set{0^{\prime},0}$ and $\Set{1^{\prime},1}$ are five.

We can prove the following.

Theorem 3.17**.**

Let $m,n\geq 1$ and $m+n\geq 3$. Then eigenvalues of $\tilde{H}_{K_{m,n}}$ are $-2m^{n-2}n^{m-2}$, $-m^{n-2}n^{m-1}$, $-m^{n-1}n^{m-2}$ and $m^{n-2}n^{m-2}(m+n-1)(m+n-2)$. The dimensions of eigenspaces of $-2m^{n-2}n^{m-2}$, $-m^{n-2}n^{m-1}$, $-m^{n-1}n^{m-2}$ and $m^{n-2}n^{m-2}(m+n-1)(m+n-2)$ are $(m-1)(n-1)$, $n-1$, $m-1$ and $1$, respectively.

Remark 3.18**.**

Theorem 3.17 implies that one of the eigenvalues of $\tilde{H}_{K_{m,n}}$ is positive and that the others are negative.

Theorem 3.17 implies Theorem 3.19.

Theorem 3.19**.**

Let $m,n\geq 1$ and $m+n\geq 3$. Then the determinant of $\tilde{H}_{K_{m,n}}$ is

[TABLE]

Hence the Hessian $\det H_{K_{m,n}}$ does not vanish.

Let us prove Theorem 3.17. We compute the eigenvalues of the Hessian matrices of $K_{X,Y}=K_{m,n}$. For $e,e^{\prime}\in E(K_{X,Y})$, we define $a(e,e^{\prime})$ by

[TABLE]

We consider a group action to $V(K_{m,n})$ as follows: Let $G$ be the cyclic group generated by $\sigma$ of order $n$. Let

[TABLE]

The action of $G$ on $V(K_{m,n})$ induces an action on $E(K_{m,n})$ by

[TABLE]

for $i^{\prime}\in X$ and $j\in Y$. We define

[TABLE]

where $e_{i}=\Set{i^{\prime},0}\in E(K_{m,n})$. Note that $C^{ij}$ is an $n\times n$ cyclic matrix by the definition of the action. We have

[TABLE]

Hence $m^{n-3}n^{m-3}C$ is $\tilde{H}_{K_{m,n}}$ by Lemma 3.15.

Let us calculate eigenvalues of $C$ by Theorem 2.5. Let $a=n(2m+n-2)$, $b=m(2n+m-2)$, $c=(m+n)(m+n-2)$, $A=aJ_{nn}+(-a)I_{n}$, and $B=cJ_{nn}+(b-c)I_{n}$. Then we have $C^{ij}=A$ if $i=j$, otherwise $C^{ij}=B$. By Lemma 2.4, we obtain the eigenvalues of $A$ and $B$. The eigenvalues of $A$ are $(n-1)a$ and $-a$. The dimensions of eigenspaces of $(n-1)a$ and $-a$ are $1$ and $n-1$, respectively. The eigenvalues of $B$ are $b+(n-1)c$ and $b-c$. The dimensions of eigenspaces of $b+(n-1)c$ and $b-c$ are $1$ and $n-1$, respectively. For $k=0$, define

[TABLE]

For $1\leq k\leq m-1$, define

[TABLE]

For $0\leq k\leq m-1$, we define

[TABLE]

Then we have

[TABLE]

for $1\leq k\leq m-1$. By Lemma 2.4, we obtain the eigenvalues of $\bar{C}^{(0)}$ and $\bar{C}^{(k)}$. The eigenvalues of $\bar{C}^{(0)}$ are

[TABLE]

and

[TABLE]

The dimensions of the eigenspaces are $1$ and $m-1$, respectively. For $1\leq k\leq m-1$, the eigenvalues of $\bar{C}^{k}$ are

[TABLE]

and

[TABLE]

The dimensions of the eigenspaces are $1$ and $m-1$, respectively. Theorem 2.5 implies Proposition 3.17.

4. The Lefschetz property for an algebra associated to a graphic matroid

In this section, we will show the Lefschetz property of the algebra associated to the graphic matroid of the complete graph and the complete bipartite graph with at most five vertices (Theorems 4.5 and 4.6).

Definition 4.1**.**

Let $A=\bigoplus_{k=0}^{s}A_{k}$, $A_{s}\neq\boldsymbol{0}$, be a graded Artinian algebra. We say that $A$ has the strong Lefschetz property if there exists an element $L\in A_{1}$ such that the multiplication map $\times L^{s-2k}\colon A_{k}\to A_{s-k}$ is bijective for all $k\leq\frac{s}{2}$. We call $L\in A_{1}$ with this property a strong Lefschetz element.

Let $\mathbb{K}$ be a field of characteristic zero. For a homogeneous polynomial $F\in\mathbb{K}[x_{1},x_{2},\ldots,x_{N}]$, we define $\operatorname{Ann}(F)$ by

[TABLE]

Then $\operatorname{Ann}(F)$ is a homogeneous ideal of $\mathbb{K}[x_{1},\ldots,x_{N}]$. We consider $A=\mathbb{K}[x_{1},\ldots,x_{N}]/\operatorname{Ann}(F)$. Since $\operatorname{Ann}(F)$ is homogeneous, the algebra $A$ is graded. Furthermore $A$ is an Artinian Gorenstein algebra. Conversely, a graded Artinian Gorenstein algebra $A$ has the presentation

[TABLE]

for some homogeneous polynomial $F\in\mathbb{K}[x_{1},x_{2},\ldots,x_{N}]$. We decompose $A$ into the homogeneous components $A_{k}$. Let $\Lambda_{k}$ be the basis for $A_{k}$. We define the matrix $H_{F}^{(k)}$ by

[TABLE]

The determinant of $H_{F}^{(k)}$ is called the $k$th Hessian of $F$ with respect to the basis $\Lambda_{k}$.

Remark 4.2**.**

We define the [math]th Hessian of $F$ is $F$.

There is a criterion for the strong Lefschetz property for a graded Artinian Gorenstein algebra.

Theorem 4.3** (Watanabe [9], Maeno–Watanabe [6]).**

Consider the graded Artinian Gorenstein algebra $A$ with the following presentation and decomposition: $A=\mathbb{K}[x_{1},x_{2},\ldots,x_{N}]/\operatorname{Ann}(F)=\bigoplus_{k=0}^{s}A_{k}$. Let $L=a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{N}x_{N}$. The multiplication map $\times L^{s-2k}\colon A_{k}\to A_{s-k}$ is bijective if and only if

[TABLE]

Definition 4.4**.**

For a graph $\Gamma$ with $N$ edges, we define the graded Artinian Gorenstein algebra $A_{M(\Gamma)}$ by

[TABLE]

If a graph $\Gamma$ has $n+1$ vertices, then the top degree of $A_{M(\Gamma)}$ is $n$.

Theorem 4.5**.**

The algebra $A_{M(K_{n})}$ has the strong Lefschetz property for $n\leq 5$. The element $x_{1}+\cdots+x_{N}$ is a strong Lefschetz element.

Proof.

Let $n\leq 5$. It follows from Theorem 3.4 that $H_{F_{K_{n}}}^{(1)}=H_{K_{n}}$, and $\det H_{F_{K_{n}}}^{(1)}(1,\ldots,1)=\det\tilde{H}_{K_{n}}\neq 0$. Hence $A_{K_{n}}$ has the strong Lefschetz property, and the element $x_{1}+\cdots+x_{N}$ is a strong Lefschetz element. ∎

Similarly Theorem 4.6 follows from Remark 4.2 and Theorem 3.19.

Theorem 4.6**.**

The algebra $A_{M(K_{m,n})}$ has the strong Lefschetz property for $m+n\leq 5$ and $m,n\geq 1$. The element $x_{1}+\cdots+x_{N}$ is a strong Lefschetz element.

Remark 4.7**.**

In [8], it is shown that the Hessian of any simple graph does not vanish. In the paper, it is shown that the Kirchhoff polynomial of a complete graph is the irreducible relative invariant of a prehomogeneous vector space. By this fact, we can compute the Hessian as polynomials of the complete graph, and compute the Hessians of any graphs.