8 Concluding Remarks and Further Work

15 the average degree of the nodes of a linear-sized induced subgraph of a k-regular Ramanujan graph is upper bounded by roughly 1 + p k ? 1, which is less than k=2 for k 7. This shows that Ramanujan graphs of degree at least 7 are extrovert graphs. Classical results 4] require the degree to be at least 15. 1. Let H be a graph of maximumdegree at most k, and a real number no smaller than 2 p k ? 1. Theorem 1 implies that, if there exists an innnite family G n of k-regular graphs containing H as an induced subgraph and such that (G n) (1 + o(1)), then 0 (M H). (M H can be deened similarly to M X .) If k ? 1 is a prime congruent to 1 modulo 4, this condition can be shown to be suucient 20]. 2. It is still an open question whether there exists a family of Ramanujan graphs with linear expansion at most k=2. 3. It would be interesting to calculate the exact value of the linear expansion of the known explicit constructions of Ramanujan graphs 23, 26]. Theorem 2 shows that it is at least k=2. On the other hand, an easy combinatorial argument shows that the linear expansion of any family of k-regular graphs is at most k?1. Besides being Ramanujan, the graphs constructed in 23, 26] have other interesting combinatorial properties. For example, they are Cayley graphs and have high girth, unlike the graphs that we constructed in Section 5. This leads us to conjecture that their linear expansion is strictly greater than k=2. Any explicit construction of k-regular graphs with provable linear expansion strictly greater than k=2 would also be interesting. 14 0 (A W) is the largest eigenvalue of A W in absolute value, the largest eigenvalue of A W l is (0 (A W)) l. Using Lemma 2 and Fact 1, we conclude that (A W l W) W (0 (A W)) l jj W jj 2 n (1 + k ? 1) l ; as desired. Deene the density of a subset of vertices to be the ratio of its size to the total number of vertices. Corollary 1 is optimal in the following sense: for any rational 2 1=2; 1] and any rational 2 0; 1], there exists an arbitrary large k-regular graph G and a subset W …