On graphs with just three distinct eigenvalues

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Let G be a graph of order n with (0, 1)-adjacency matrix A. An eigenvalue σ of A is said to be an eigenvalue of G, and σ is a main eigenvalue if the eigenspace EA(σ) is not orthogonal to the all-1 vector in IR. Always the largest eigenvalue, or index, of G is a main eigenvalue, and it is the only main eigenvalue if and only if G is regular. We say that G is an integral graph if every eigenvalue...

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We consider undirected non-regular connected graphs without loops and multiple edges (other than complete bipartite graphs) which have exactly three distinct eigenvalues (such graphs are called non-standard graphs). The interest in these graphs is motivated by the questions posed by W. Haemers during the 15th British Combinatorial Conference (Stirling, July 1995); the main question concerned th...

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Let Dv,b,k denote the family of all connected block designs with v treatments and b blocks of size k. Let d ∈ Dv,b,k. The replication of a treatment is the number of times it appears in the blocks of d. The matrix C(d) = R(d) − 1 kN(d)N(d) > is called the information matrix of d where N(d) is the incidence matrix of d and R(d) is a diagonal matrix of the replications. Since d is connected, C(d)...

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ژورنال

عنوان ژورنال: Linear Algebra and its Applications

سال: 2016

ISSN: 0024-3795

DOI: 10.1016/j.laa.2016.06.031