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February 2020 On quadratic embedding constants of star product graphs
Wojciech MŁOTKOWSKI, Nobuaki OBATA
Hokkaido Math. J. 49(1): 129-163 (February 2020). DOI: 10.14492/hokmj/1591085015

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 $G=G_1\star\cdots \star G_r$ 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.

Citation

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Wojciech MŁOTKOWSKI. Nobuaki OBATA. "On quadratic embedding constants of star product graphs." Hokkaido Math. J. 49 (1) 129 - 163, February 2020. https://doi.org/10.14492/hokmj/1591085015

Information

Published: February 2020
First available in Project Euclid: 2 June 2020

zbMATH: 07209523
MathSciNet: MR4105539
Digital Object Identifier: 10.14492/hokmj/1591085015

Subjects:
Primary: 05C50
Secondary: 05C12, 05C76

Rights: Copyright © 2020 Hokkaido University, Department of Mathematics

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Vol.49 • No. 1 • February 2020
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