A Generalized Alon-Boppana Bound and Weak Ramanujan Graphs

A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs. In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph $G$ with diameter $k$ and vertex set $V$, the smallest nontrivial eigenvalue $\lambda_1$ of the normalized Laplacian $\mathcal L$ satisfies$$ \lambda_1 \leq 1-\sigma \big(1- \frac c {k} \big)$$ for some constant $c$ where $\sigma = 2\sum_v d_v \sqrt{d_v-1}/\sum_v d_v^2 $ and $d_v$ denotes the degree of the vertex $v$.We consider weak Ramanujan graphs defined as graphs satisfying $ \lambda_1 \geq 1-\sigma$. We examine the vertex expansion and edge expansion of weak Ramanujan graphs and then use the expansion properties among other methods to derive the above Alon-Boppana bound. A corrigendum was added on the 3rd of November 2017.

Cubical coloring — fractional covering by cuts and semidefinite programming

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).