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Results tagged with spectral-graph-theory
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Questions related to the spectrum of graphs, defined using one of the possible variants of the discrete Laplace operator or Laplacian matrix. See https://en.wikipedia.org/wiki/Discrete_Laplace_operator
20
votes
Accepted
What is a "Ramanujan Graph"?
Ramanujan graphs were first defined by Lubotzky, Phillips and Sarnak:
http://math1.math.huji.ac.il/~alexlub/PAPERS/ramanujan%20graphs/ramanujanGraphs.pdf
As you can see, they are $d$-regular and and …
- 11.1k
7
votes
About the roots of the matching polynomial
I really like that paper:
C. D. Godsil and I. Gutman. On the matching polynomial of a graph. In L. Lovasz and V. T. Sos, editors, Algebraic Methods in graph theory, volume I of Colloquia Mathematica …
- 11.1k
1
vote
How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?
For the Cayley graph of $SL_2(\mathbb{F}_p)$ with respect to $(1,2,0,1)$, its transpose, and their inverses (so a 4-regular graph), there is a short and elegant proof of Margulis that the girth is log …
- 11.1k
1
vote
Extreme Laplacian eigenvalues
Let me turn the previous remarks into a tentative answer. Fix distinct primes $p,q$ and consider the Ramanujan graphs $X^{p,q}$ of Lubotzky-Phillips-Sarnak [A. Lubotzky, R. Phillips, P. Sarnak (1988). …
- 11.1k
1
vote
Spectral gap for random bipartite regular graphs
In the paper "On the second eigenvalue and random walks in random d-regular graphs" (Combinatorica 11 (1991), no. 4, 331–362), Joel Friedman considers a model of $2d$-regular random graphs on $n$ vert …
- 11.1k