Ramanujan graphs with small girth Academic Article uri icon

abstract

  • Lubotzky, in his book [9, Question 10.7.1], poses the question of clarifying the connection between the Ramanujan property and the girth. There are some theorems showing a correlation between the eigenvalue distribution and the existence of small circuits, but they are all rather weak. For ex- ample Greenberg (see [5],[9, theorem 4.5.7]) proves that an infinite family of Ramanujan graphs ...Xn → Xn−1 → ... → X1 covering each other, with Xn → X1 a regular covering map, must satisfy girth(Xn) → ∞ (in other words an example like the one given in theorem (1.2) below is not possible when the covering maps Xn →X1 are assumed to be regular). In the other direction McKay (see [10]) shows that infinite families of (q + 1)−regular graphs with asymptotically few circuits have most of their eigenvalues con- centrated in the Ramanujan interval [−2 √ q,2 √ q] … In this paper we give the following example, proving that the Ramanujan property does not …

publication date

  • January 1, 2003