The quadratic embedding constant 
of a finite simple connected
graph 
on 
vertices is defined as the maximum of the quadratic
form 
over all real 
-vectors 
satisfying 
and 
, where 
is the graph distance
matrix (Obata and Zakiyyah 2018, Choudhury and Nandi 2023, Obata 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 incorrect). For the
singleton graph, define 
by convention (Obata and Zakiyyah 2018, p. 41).
Quadratic embedding constants for the singleton graph 
(Obata and Zakiyyah 2018), complete graphs 
with 
(Obata and Zakiyyah 2018, Obata 2023), complete
bipartite graphs 
(Obata and Zakiyyah 2018, Obata 2023), cycle
graphs 
(Obata and Zakiyyah 2018, Obata 2023), path graphs 
with 
(Młotkowski 2022, Obata 2023), and wheel graphs
(E. Weisstein, Jul. 3, 2023) are given below, with 
for cycle graphs, 
for wheel graphs, and 
for complete
bipartite graphs:
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |
(7)
|
Obata (2023) gives quadratic embedding constants for complete 
-partite graphs in general.
Any graph obtained by deleting two or more edges with no shared vertices from a complete
graph has quadratic embedding constant equal to 0 (Obata and Zakiyyah 2018, p. 46).
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 two or more quadratically
embeddable graphs 
, 
, ..., each having two or more vertices,
is 
(Obata and Zakiyyah 2018, p. 44).
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.
