Quadratic Embedding Constants of Corona Graphs
Ferdi, Edy Tri Baskoro, Nobuaki Obata, Aditya Purwa Santika
TL;DR
The paper studies the quadratic embedding constant (QEC) for corona graphs G⊙H and derives a formula linking QEC(G⊙H) to QEC(G) through the H-specific ψ-function, under spectral assumptions on H.Central to the method are two analytic constructs, the ω_A and ψ_A functions, built from the adjacency spectrum of a symmetric matrix A, which encode main eigenvalues and govern theCursor's inverse branch ψ_{H*}^{−1} used in the corona formula.A general result shows QEC(G⊙H) = ψ_{H*}^{−1}(QEC(G)) when certain spectral conditions hold; a correction term γ3 captures cases where det(A_H+2+λ)=0 and can drive the value if needed, with a simpler explicit form available when H is regular with min eigenvalue −2.The paper provides explicit formulas for ω_H, ψ_H and their inverses in regular-H scenarios, and supplies a suite of examples (e.g., G⊙bar{K}_n, G⊙pK_q, G⊙pC_q) to illustrate the applicability and limitations of the main formula.
Abstract
The quadratic embedding constant (QEC) of a connected graph is defined to be the maximum of the quadratic function associated with its distance matrix on a certain unit sphere of codimension two. In this paper we derive a formula for the QEC of a corona graph $G\odot H$. It is shown that $\mathrm{QEC}(G\odot H)=ψ_{H*}^{-1}(\mathrm{QEC}(G))$ holds under some spectral assumptions on $H$, where $ψ_{H*}^{-1}$ is the inverse function of the most right branch of the analytic function $ψ_H$ defined by means of the main eigenvalues of the adjacency matrix of $H$. Moreover, if $H$ is a regular graph of which the adjacency matrix has the smallest eigenvalue $-2$, then the formula is written down explicitly.