Pairwise comparison

Signed graph

Spectrum of a matrix

Quantum mechanics

Context (archaeology)

Extension (predicate logic)

Biology

Discrete mathematics

Statistics

Differential equation

Indifference graph

Eigenvalues and eigenvectors

Computer science

Paleontology

Spectrum (functional analysis)

Physics

Matrix differential equation

Upper and lower bounds

Graph

Mathematics

Mathematical analysis

Geometry

Combinatorics

Programming language

Chordal graph

Euclidean space

Euclidean geometry

Applied Mathematics and Computation

In this paper we study connected signed graphs with 2 eigenvalues from several (theoretical and computational) perspectives. We give some basic results concerning the eigenvalues and cyclic structure of such signed graphs; in particular, we complete the list of those that are 3 or 4-regular. There is a natural relation between signed graphs and systems of lines in a Euclidean space that are pairwise orthogonal or at fixed angle, with a special role of those with 2 eigenvalues. In this context we derive a relative bound for the number of such lines (an extension of the similar bound related to unsigned graphs). We also determine all such graphs whose negative eigenvalue in not less than −2, except for so-called exceptional signed graphs. Using the computer search, we determine those with at most 10 vertices. Several constructions are given and the possible spectra of those with at most 30 vertices are listed.

Spectra of signed graphs with two eigenvalues