scholarly journals Lattices From Tight Frames and Vertex Transitive Graphs

10.37236/8502 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Lenny Fukshansky ◽  
Deanna Needell ◽  
Josiah Park ◽  
Yuxin Xin
Keyword(s):  
Tight Frames ◽  
Complete Graphs ◽  
Irreducible Group ◽  
Product Graphs ◽  
Potential Applications ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs ◽  
Root Lattices

We show that real tight frames that generate lattices must be rational. In the case of irreducible group frames, we show that the corresponding lattice is always strongly eutactic. We use this observation to describe a construction of strongly eutactic lattices from vertex transitive graphs. We show that such lattices exist in arbitrarily large dimensions and discuss some examples arising from complete graphs, product graphs, as well as some other notable examples of graphs. In particular, some well-known root lattices and those related to them can be recovered this way. We discuss various properties of this construction and also mention some potential applications of lattices generated by incoherent systems of vectors.

Trivalent Non-symmetric Vertex-Transitive Graphs of Order at Most 150

2008 ◽  
Vol 15 (03) ◽  
pp. 379-390 ◽  
Author(s):  
Xuesong Ma ◽  
Ruji Wang
Keyword(s):  
Automorphism Group ◽  
Bipartite Graphs ◽  
Type Ii ◽  
Symmetric Graph ◽  
Trivalent Graph ◽  
Block Graphs ◽  
Group A ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

Let X be a simple undirected connected trivalent graph. Then X is said to be a trivalent non-symmetric graph of type (II) if its automorphism group A = Aut (X) acts transitively on the vertices and the vertex-stabilizer Av of any vertex v has two orbits on the neighborhood of v. In this paper, such graphs of order at most 150 with the basic cycles of prime length are investigated, and a classification is given for such graphs which are non-Cayley graphs, whose block graphs induced by the basic cycles are non-bipartite graphs.

Properties of Classes of Random Graphs

1994 ◽  
Vol 3 (4) ◽  
pp. 435-454 ◽  
Author(s):  
Neal Brand ◽  
Steve Jackson
Keyword(s):  
Random Graphs ◽  
Higher Order ◽  
Order Property ◽  
First Order ◽  
New Class ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Almost All ◽  
Transitive Graphs

In [11] it is shown that the theory of almost all graphs is first-order complete. Furthermore, in [3] a collection of first-order axioms are given from which any first-order property or its negation can be deduced. Here we show that almost all Steinhaus graphs satisfy the axioms of almost all graphs and conclude that a first-order property is true for almost all graphs if and only if it is true for almost all Steinhaus graphs. We also show that certain classes of subgraphs of vertex transitive graphs are first-order complete. Finally, we give a new class of higher-order axioms from which it follows that large subgraphs of specified type exist in almost all graphs.

scholarly journals Cubic vertex-transitive graphs on up to 1280 vertices

2013 ◽  
Vol 50 ◽  
pp. 465-477 ◽  
Author(s):  
Primož Potočnik ◽  
Pablo Spiga ◽  
Gabriel Verret
Keyword(s):  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs
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