Regular graphs with maximal energy per vertex

We study the energy per vertex in regular graphs. For every k, we give an upper bound for the energy per vertex of a k-regular graph, and show that a graph attains the upper bound if and only if it is the disjoint union of incidence graphs of projective planes of order k-1 or, in case k=2, the disjoint union of triangles and hexagons. For every k, we also construct k-regular subgraphs of incidence graphs of projective planes for which the energy per vertex is close to the upper bound. In this way, we show that this upper bound is asymptotically tight

Families of Small Regular Graphs of Girth 5

In this paper we obtain $(q+3)$--regular graphs of girth 5 with fewer vertices than previously known ones for $q=13,17,19$ and for any prime $q \ge 23$ performing operations of reductions and amalgams on the Levi graph $B_q$ of an elliptic semiplane of type ${\cal C}$. We also obtain a 13-regular graph of girth 5 on 236 vertices from $B_{11}$ using the same technique

Scalar Field Dark Matter: head-on interaction between two structures

In this manuscript we track the evolution of a system consisting of two self-gravitating virialized objects made of a scalar field in the newtonian limit. The Schr\"odinger-Poisson system contains a potential with self-interaction of the Gross-Pitaevskii type for Bose Condensates. Our results indicate that solitonic behavior is allowed in the scalar field dark matter model when the total energy of the system is positive, that is, the two blobs pass through each other as should happen for solitons; on the other hand, there is a true collision of the two blobs when the total energy is negative.Comment: 8 revtex pages, 11 eps figures. v2 matches the published version. v2=v1+ref+minor_change

Quadratic vector fields with a weak focus of third order

We study phase portraits of quadratic vector fields with a weak focus of third order at the origin. We show numerically the existence of at least 20 different global phase portraits for such vector fields coming from exactly 16 different local phase portraits available for these vector fields. Among these 20 phase portraits, 17 have no limit cycles and three have at least one limit cycle