The quadratic embedding constant of a finite simple connected
graph
on
vertices is defined as the maximum of the product over all real -vectors satisfying and , where is the graph distance
matrix (Obata and Zakiyyah 2018, Obata 2022, Choudhury and Nandi 2023). Obata
and Zakiyyah (2018) give quadratic embedding constants for connected
graphs on 5 and fewer vertices (though the value given for the 12th graph on
5 nodes, i.e., the kite graph, is incorect).

Quadratic embedding constants for the singleton graph
(Obata and Zakiyyah 2018), complete graphs
with
(Obata and Zakiyyah 2018, Obata 2022), complete
bipartite graphs (Obata and Zakiyyah 2018, Obata 2022), cycle
graphs
(Obata and Zakiyyah 2018, Obata 2022), path graphs with (Młotkowski 2022, Obata 2022), and wheel
graphs (E. Weisstein, Jul. 3, 2023) are given by

Obata (2022) gives quadratic embedding constants for complete -partite graphs in general.

Any graph obtained by deleting two or more disjoint subsets from a complete graph has quadratic embedding constant equal to 0 (Obata and Zakiyyah 2018).
This includes the square graph , wheel graph , octahedral graph ,
queen graph , 16-cell graph ,
and complete -partite graphs , , etc.

The quadratic embedding constant of a graph Cartesian product of graphs , , ... each having two or more vertices has (Obata 2022).

For a connected graph whose graph distance matrix
has equal row sums, the quadratic embedding constant is given by the second largest
eigenvalue of (Obata and Zakiyyah 2018).

A connected graph for which is called a quadratically
embeddable graph.

## See also

Graph Distance Matrix,

Quadratically Embeddable Graph
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## References

Choudhury, P. N. and Nandi, R. "Quadratic Embedding Constants of Graphs: Bounds and Distance Spectra." 27 Jun 2023. https://arxiv.org/abs/2306.15589.Młotkowski,
W. "Quadratic Embedding Constants of Path Graphs." *Linear Algebra Appl.* **644**,
95-107, 2022.Obata, N. "Complete Multipartite Graphs of Non-QE
Class." 12 Jun 2022. https://arxiv.org/abs/2206.05848.Obata,
N. and Zakiyyah, A. Y. "Distance Matrices and Quadratic Embedding of Graphs."
*Elec. J. Graph Th. Appl.* **6**, 37-60, 2018.Schoenberg, I. J.
"Metric Spaces and Positive Definite Functions." *Trans. Amer. Math.
Soc.* **44**, 522-536, 1938.
## Cite this as:

Weisstein, Eric W. "Quadratic Embedding Constant."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/QuadraticEmbeddingConstant.html