On a family of strongly regular graphs withλ=1

Abstract

In this paper, we give a complete description of strongly regular graphs with parameters ((n 2 + 3n − 1) 2 , n 2 (n + 3), 1, n(n + 1)).All possible such graphs are: the lattice graph L 3,3 with parameters (9, 4, 1, 2), the Brouwer-Haemers graph with parameters (81, 20,1,6), and the Games graph with parameters (729, 112,1, 20).

On a Family of Diamond-Free Strongly Regular Graphs

“…It follows from the orbit-stabilizer theorem that n 2 + 3n − 2 = 2 t , for some integer t. We have (2n + 3) 2 = 2 t+2 + 17. Using a result in [2, p. 401], we obtain that (n, t) ∈ {(1, 1), (2, 3), (3,4), (10, 7)}. Since n 3, we conclude that (n, ν) ∈ {(3, 256), (10, 16384)}.…”

“…IV. Partial Quadrangle PQ (3,35,20) In the following, we demonstrate that there exists no PQ (3,35,20), or equivalently, there is no diamond-free SRG(676, 108, 2, 20). Notice that this strongly regular graph belongs to the family (2) with n = 4 and λ = 2.…”

“…Recently, Bondarenko and Radchenko showed in [3] that a PQ(2, (n 3 + 3n 2 − 2)/2, n 2 + n), or equivalently, an SRG((n 2 + 3n − 1) 2 , n 2 (n + 3), 1, n(n + 1)), exists if and only if n ∈ {1, 2, 4}. Let S be a PQ(3, (n + 3)(n 2 − 1)/3, n 2 + n) such that for every two non-collinear points p 1 and p 2 , there is a point q non-collinear with p 1 , p 2 , and all points collinear with both p 1 and p 2 .…”

Using Symbolic Computation to Prove Nonexistence of Distance-Regular Graphs

“…Lemma 6. Let Σ be a distance-regular graph with intersection array (7), and u, v, w be vertices of Σ that are pairwise at distance 2. Then u v w 1 1 1 ≤ 1.…”

Spectra of some Operations on Graphs

“…The related matrix -the adjacency matrix of a graph and its eigenvalues were much more investigated in the past than the Laplacian matrix. In the same time, the Laplacian spectrum is much more natural and more important than the adjacency matrix spectrum because of it numerous application in mathematical physics, chemistry and financial mathematics (see papers [1,3,4,6,7,8]).…”