Quadratic embedding constants of fan graphs and graph joins

TL;DR

This work derives an explicit formula for the quadratic embedding constant (QE constant) of graph joins, focusing on $\mathrm{QEC}(\bar{K}_m+G)$ through a Lagrange-multiplier framework that reduces to a minimal $\alpha$-value, yielding $\mathrm{QEC}(G_1+G_2)=-\tilde{\alpha}-2$. Specializing to fan graphs $K_1+P_n$, the authors define a polynomial $\Phi_n(x)$ via compressed Chebyshev polynomials and show $\mathrm{QEC}(K_1+P_n)=-\tilde{\alpha}_n-2$, where $\tilde{\alpha}_n$ is the minimal root of $\Phi_n$. For even $n$, $\tilde{\alpha}_n$ coincides with the path-adjacency minimum eigenvalue $\min\mathrm{ev}(A_n)=-2\cos(\pi/(n+1))$, giving a closed form $\mathrm{QEC}(K_1+P_n)=-4\sin^2(\pi/(2(n+1)))$, while for odd $n$ one has $\min\mathrm{ev}(A_{n+1})\le\tilde{\alpha}_n<\min\mathrm{ev}(A_n)$. The results connect QE constants to Chebyshev polynomials, reveal a decomposition involving partial Chebyshev polynomials, and provide explicit embeddings in Euclidean space for certain fan graphs. These findings advance a quantitative understanding of Euclidean distance realizations of graph joins and yield exact QE constants for a fundamental family of graphs.

Abstract

We derive a general formula for the quadratic embedding constant of a graph join $\bar{K}_m+G$, where $\bar{K}_m$ is the empty graph on $m\ge1$ vertices and $G$ is an arbitrary graph. Applying our formula to a fan graph $K_1+P_n$, where $K_1=\bar{K}_1$ is the singleton graph and $P_n$ is the path on $n\ge1$ vertices, we show that $\mathrm{QEC}(K_1+P_n)=-\tildeα_n-2$, where $\tildeα_n$ is the minimal zero of a new polynomial $Φ_n(x)$ related to Chebyshev polynomials of the second kind. Moreover, for an even $n$ we have $\tildeα_n=\min\mathrm{ev}(A_n)$, where the right-hand side is the An minimal eigenvalue of the adjacency matrix $A_n$ of $P_n$. For an odd $n$ we show that $\min\mathrm{ev}(A_{n+1})\le\tildeα_n<\min\mathrm{ev}(A_n)$.

Theorems & Definitions (60)

• Theorem 2.1
• Remark 2.2
• Proposition 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof