[Paper Review] On Quadratic Embedding Constants of Star Product Graphs

Wojciech Młotkowski, Nobuaki Obata|arXiv (Cornell University)|Feb 4, 2018

Advanced Combinatorial Mathematics4 references3 citations

TL;DR

This paper derives an implicit formula for the quadratic embedding constant (QEC) of star product graphs $G = G_1 \star \cdots \star G_r$, establishing a connection between the QEC of the component graphs and the solution of a specific algebraic equation. The key result is that $\mathrm{QEC}(\mathbb{Z}) = \mathrm{QEC}(\mathbb{Z}_+) = -\frac{1}{2}$, obtained via a limit argument on finite path graphs $P_n$, and a new integer sequence arises from combinatorial identities used in the analysis.

ABSTRACT

A connected graph $G$ is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant $\mathrm{QEC}(G)$ is non-positive. For a finite star product of (finite or infinite) graphs $G=G_1\star\dotsb \star G_r$ an estimate of $\mathrm{QEC}(G)$ is obtained after a detailed analysis of the minimal solution of a certain algebraic equation. For the path graph $P_n$ an implicit formula for $\mathrm{QEC}(P_n)$ is derived, and by limit argument $\mathrm{QEC}(\mathbb{Z})=\mathrm{QEC}(\mathbb{Z}_+)=-1/2$ is shown. During the discussion a new integer sequence is found.

Motivation & Objective

• To establish a quantitative estimate for the quadratic embedding constant (QEC) of star product graphs $G_1 \star \cdots \star G_r$.
• To analyze the minimal solution of an algebraic equation involving parameters derived from component graphs.
• To derive an implicit formula for $\mathrm{QEC}(P_n)$ for finite path graphs $P_n$.
• To compute the QEC for infinite path graphs $\mathbb{Z}_+$ and $\mathbb{Z}$ via a limit process.

Proposed method

• Analyzes the minimal solution of the algebraic equation $\sum_{j=1}^{r}\frac{d_j}{a_j d_j + a_j - \lambda} = \frac{1}{\lambda}$, where $a_j$ and $d_j$ are parameters derived from component graphs.
• Connects the conditional minimum of a quadratic form $\phi(x_0, \mathbf{x}_1, \dots, \mathbf{x}_r)$ under constraints to the minimal solution of the algebraic equation.
• Uses spectral and variational techniques to relate the QEC of the star product to the solution of the algebraic equation.
• Applies limit arguments to finite path graphs $P_n$ to compute $\mathrm{QEC}(\mathbb{Z}_+)$ and $\mathrm{QEC}(\mathbb{Z})$.
• Derives combinatorial identities involving sums of $\min\{i,j\}$, $i(n-i)$, and alternating signs to estimate $\mathrm{QEC}(P_n)$.
• Identifies a new integer sequence $a_n$ from the combinatorial sum $\sum_{i,j=1}^n \min\{i,j\} \, i(n-j)j(n-j)(-1)^{i+j}$, shown to be the convolution of $\lceil n^2/2 \rceil$ with itself.

Experimental results

Research questions

• RQ1 What is the quadratic embedding constant (QEC) of a star product graph $G_1 \star \cdots \star G_r$ in terms of the QECs of its components?
• RQ2 Can an implicit formula for $\mathrm{QEC}(P_n)$ be derived for finite path graphs $P_n$?
• RQ3 What is the value of $\mathrm{QEC}(\mathbb{Z}_+)$ and $\mathrm{QEC}(\mathbb{Z})$ for the one- and two-sided infinite paths?
• RQ4 How are the combinatorial identities used in estimating $\mathrm{QEC}(P_n)$ related to known integer sequences?
• RQ5 Is there a new integer sequence arising from the combinatorial structure of the QEC estimation for path graphs?

Key findings

• The QEC of the infinite path graph $\mathbb{Z}_+$ is $\mathrm{QEC}(\mathbb{Z}_+) = -\frac{1}{2}$, derived as the limit of $\mathrm{QEC}(P_n)$ as $n \to \infty$.
• The QEC of the two-sided infinite path $\mathbb{Z}$ is $\mathrm{QEC}(\mathbb{Z}) = -\frac{1}{2}$, established via the same limit argument.
• An implicit formula for $\mathrm{QEC}(P_n)$ is derived using the minimal solution of the algebraic equation $\sum_{j=1}^{r}\frac{d_j}{a_j d_j + a_j - \lambda} = \frac{1}{\lambda}$.
• A new integer sequence $a_n = \frac{n}{240}\{2n^4 + 20n^2 - 7 + 15(-1)^n\}$ arises from the combinatorial sum in the QEC estimation for $P_n$.
• The sequence $a_n$ is shown to be the convolution of the sequence $b_n = \lceil n^2/2 \rceil$ with itself, and its generating function is $\sum_{n=0}^\infty a_n z^n = \frac{z^2(1+z^2)^2}{(1+z)^2(1-z)^6}$.
• The conditional minimum of a quadratic form under constraints is proven to coincide with the minimal solution of the key algebraic equation, providing a variational characterization of the QEC of star product graphs.