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By Rudenskaya O. G.

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The proof actually shows that if we maximize the left value and minimize the right value over x ≥ 0, then we get the same value from both cases. If we let j be the all-1 vector and A be the adjacency matrix of a graph G with degrees d1 , d2 , . . , dn , then we get min di ≤ λ1 ≤ max di . 5 with the same vector j, we get n1 λ. This results in the following theorem. 8 If G is a graph with degrees d1 , d2 , . . , dn and maximum eigenvalue λ1 , then 1 n n di ≤ λ1 ≤ max di . i=1 i Equality is attained if and only if the graph is regular.

Xn ) be defined by letting x j = εk . 40 Michael Doob Then (Ax) j = x j−1 + x j+1 (subscripts taken modulo n), and x j−1 + x j+1 e2iπ( j−1)k/n + e2iπ( j+1)k/n = = e−2iπk/n + e2iπ k/n = εk−1 + εk . xj e2iπ jk/n Hence, Ax = (εk−1 + εk )x and, on letting k range from 1 to n, we obtain all the eigenvalues of Cn . Notice that the roots of unity εk and εn−k yield the same eigenvalue, and so, unless k = n or n/2, this eigenvalue has multiplicity 2; otherwise, the multiplicity is 1. Similar labellings can be used to find the eigenvalues of a path.

We define n i (v) to be the number of vertices at distance i from v, and for two vertices v and w at distance k from each other, pikj (v, w) is the number of vertices at distance i from v and distance j from w. If n i (v) is the same for all v, and if pikj (v, w) is the same for all vertices v and w at distance k from each other, then the graph is distance-regular. A strongly regular graph with parameters (n, r, λ, µ) is such a graph with n 1 (v) = r, n 2 (v) = 1 2 = λ and p11 = µ. The d-dimensional cube Q d is distance-regular n − r − 1, p11 with diameter d.

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3-Quasiperiodic functions on graphs and hypergraphs by Rudenskaya O. G.

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