# Graphs from Generalized Kac-Moody Algebras

@article{Terlep2012GraphsFG,
title={Graphs from Generalized Kac-Moody Algebras},
author={T. Arthur Terlep and Jason S. Williford},
journal={SIAM J. Discret. Math.},
year={2012},
volume={26},
pages={1112-1120}
}
• Published 16 August 2012
• Mathematics
• SIAM J. Discret. Math.
In this paper, we construct new families of graphs whose automorphism groups are transitive on 3-paths. These graphs are constructed from certain Lie algebras related to generalized Kac--Moody algebras of rank two. We will show that one particular subfamily gives new lower bounds on the number of edges in extremal graphs with no cycles of length fourteen.
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## References

SHOWING 1-10 OF 30 REFERENCES
Ramanujan graphs of very large girth based on octonions
• Mathematics
• 2010
We present a generalization of the construction of graphs by Lubotzky, Phillips and Sarnak in their celebrated article "Ramanujan graphs". The new approach consists in using octonion algebras rather
Introduction to Lie Algebras and Representation Theory
Preface.- Basic Concepts.- Semisimple Lie Algebras.- Root Systems.- Isomorphism and Conjugacy Theorems.- Existence Theorem.- Representation Theory.- Chevalley Algebras and Groups.- References.-
General Properties of Some Graphs Defined by Systems of Equations
• Mathematics
Electron. Notes Discret. Math.
• 2000
New Examples of Graphs without Small Cycles and of Large Size
• Mathematics
Eur. J. Comb.
• 1993
A new infinite series of bipartite q-regular edge-transitive graphs of order 2q5 and girth 10 is constructed, motivated by some results on embeddings of Chevalley group geometries in the corresponding Lie algebras and a construction of a blow-up for an incident system and a graph.
On Arithmetic Progressions of Cycle Lengths in Graphs
This paper proves that, for k > 2, a bipartite graph of average degree at least 4k and girth g contains cycles of (g/2 − 1)k consecutive even lengths.
On generalizing generalized polygons
The purpose of this paper is to reveal in geometric terms a decade-old construction of certain families of graphs with nice extremal properties. Construction of the graphs in question is motivated by
Ramanujan Graphs
In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication
Extremal graphs with no C4's, C6's, or C10's
• R. Wenger
• Mathematics
J. Comb. Theory, Ser. B
• 1991