In this paper, triangle-free distance-regular graphs with diameter 3 and an eigenvalue θ with multiplicity equal to their valency are studied. Let Γ be such a graph. We first show that θ = −1 if and… (More)

LetΓ be a distance-regular graphwith valency k ≥ 3 and diameter d ≥ 2. It is well known that the Schur product E ◦ F of any two minimal idempotents of Γ is a linear combination of minimal idempotents… (More)

Several methods for representing (drawing) graphs are considered, which minimize different energies of the representations. The idea of representations is generalized to maps. Some well-known… (More)

In the paper we show that all combinatorial triangle-free configurations for v ≤ 18 are geometrically realizable. We also show that there is a unique smallest astral (183) triangle-free configuration… (More)

It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements,… (More)