We determine all graphs whose adjacency matrix has at most two eigenvalues (multiplicities included) different from ±1 and decide which of these graphs are determined by their spectrum. This includes the so-called friendship graphs, which consist of a number of edge-disjoint triangles meeting in one vertex. It turns out that the friendship graph is determined by its spectrum, except when the number of triangles equals sixteen.Computing det(Q + I) and det(Q − I) shows that Q has no eigenvalue −1, and Q has an eigenvalue 1 if and only if b = 2. In case b = 2 we can rewrite A aswith k ≥ 2. Thus we obtained the graphs of Case (ii). If ΓX has no edges and at least two vertices, then these two vertices have the same neighbors, contradiction. So |ΓX| = 1 and we find

From Alon and Boppana, and Serre, we know that for any given integer k ≥ 3 and real number λ < 2 √ k − 1, there are only finitely many k-regular graphs whose second largest eigenvalue is at most λ. In this paper, we investigate the largest number of vertices of such graphs.values of graphs, see [9,18].The second eigenvalue of a regular graph is a parameter of interest in the study of graph connectivity and expanders (see [1,9,24] for example). In this paper, we investigate the maximum order v(k, λ) of a connected k-regular graph whose second largest eigenvalue is at most some given parameter λ. As a consequence of work of Alon and Boppana, and of Serre [1,12,16,24,25,28,31,35,36,42], we know that v(k, λ) is finite for λ < 2 √ k − 1. The recent result of Marcus, Spielman and Srivastava [29] showing the existence of infinite families of Ramanujan graphs of any degree at least 3 implies that v(k, λ) is infinite for λ ≥ 2 √ k − 1. For any λ < 0, the parameter v(k, λ) can be determined using the fact that a graph with only one nonnegative eigenvalue is a complete graph. Indeed, if a graph has only one nonnegative eigenvalue, then it must be connected. If a connected graph G is not a complete graph, then G contains an induced subgraph isomorphic to K 1,2 , so Cauchy eigenvalue interlacing (see [9, Proposition 3.2.1]) implies λ 2 (G) ≥ λ 2 (K 1,2 ) = 0, contradiction. Thus v(k, λ) = k +1 for any λ < 0 and the unique graph meeting this bound is K k+1 . The parameter v(k, 0) can be determined using the fact that a graph with exactly one positive eigenvalue must be a complete multipartite graph (see [7, page 89]). The largest k-regular complete multipartite graph is the complete bipartite graph K k,k , since a k-regular t-partite graph has tk/(t − 1) vertices. Thus v(k, 0) = 2k, and K k,k is the unique graph meeting this bound. The values of v(k, −1) and v(k, 0) also follow from Theorem 2.3 in Section 2 below.Results from Bussemaker, Cvetković and Seidel [10] and Cameron, Goethals, Seidel, and Shult [11] give a characterization of the regular graphs with smallest eigenvalue λ min ≥ −2.Since the second eigenvalue of the complement of a regular graph is λ 2 = −1 − λ min , the regular graphs with second eigenvalue λ 2 ≤ 1 are also characterized. This characterization can be used to find v(k, 1) (see Section 3).The values remaining to be investigated are v(k, λ) for 1 < λ < 2 √ k − 1. The parameter v(k, λ) has been studied by Teranishi and Yasuno [44] and Høholdt and Justesen [22] for the class of bipartite graphs in connection with problems in design theory, finite geometry and coding theory. Some results involving v(k, λ) were obtained by Koledin and Staníc [26,27,43] and Richey, Shutty and Stover [47] who implemented Serre's quantitative version of the Alon-Boppana Theorem [42] to obtain upper bounds for v(k, λ) for several values of k and λ. For certain values of k and λ, Richey, Shutty and Stover [47] made some conjectures about v(k, λ). We will prove some of their conjectures and disprove others in this paper. Reingold...

The simplicial rook graph SR(m, n) is the graph of which the vertices are the sequences of nonnegative integers of length m summing to n, where two such sequences are adjacent when they differ in precisely two places. We show that SR(m, n) has integral eigenvalues, and smallest eigenvalue s = max −n, − m 2 , and that this graph has a large part of its spectrum in common with the Johnson graph J (m + n − 1, n). We determine the automorphism group and several other properties.

We determine all graphs for which the adjacency matrix has at most two eigenvalues (multiplicities included) not equal to −2, or 0, and determine which of these graphs are determined by their adjacency spectrum. Keywords Graph spectrum • Spectral characterizations • Adjacency matrix Mathematics Subject Classification 05C50 Dedicated to Andries E. Brouwer on the occasion of his 65th birthday. This is one of several papers published in Designs, Codes and Cryptography comprising the special issue in honor of Andries Brouwer's 65th birthday.

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