Primary Non-QE Graphs on Six Vertices

Abstract

A connected graph is called of non-QE class if it does not admit a quadratic embedding in a Euclidean space. A non-QE graph is called primary if it does not contain a non-QE graph as an isometrically embedded proper subgraph. The graphs on six vertices are completely classified into the classes of QE graphs, of non-QE graphs, and of primary non-QE graphs.

Complete multipartite graphs of non-QE class

“…In Section 4 we determine all complete multipartite graphs of non-QE class (Theorem 1.2) and specify primary ones (Theorem 1.3). All primary non-QE graphs on six or fewer vertices are already determined [17]. As a result, we find three new primary non-QE graphs on seven vertices.…”

“…On the other hand, as is mentioned in Example 2.6, there are exactly two primary non-QE graphs on five vertices. Moreover, it is proved [17] that there are exactly three primary non-QE graphs on six vertices. They are the graphs G6-30, G6-60 and G6-84, see In this line an interesting question is to systematically construct primary non-QE graphs and to study the distribution of their QE constants.…”

“…An advantage of the QE constant is that (1) is determined explicitly or estimated finely by means of the method of Lagrange's multipliers. The QE constants are explicitly known for some special series of graphs, see also [4,5,12,14,15,17,19].…”