Regular graphs with maximal energy per vertex

Regular graphs with maximal energy per vertex

Regular vertex diameter critical graphs

A graph is called vertex diameter critical if its diameter increases when any vertex is removed. Regular vertex diameter critical graphs of every valency k ≥ 2 and diameter d ≥ 2 exist, raising the question of identifying the smallest such graphs. We describe an infinite family of k-regular vertex diameter critical graphs of diameter d with at most kd+ (2k − 3) vertices. This improves the previ...

Unicyclic graphs with maximal energy

Let G be a graph on n vertices and let λ1, λ2, . . . , λn be its eigenvalues. The energy of G is defined as E(G) = |λ1| + |λ2| + · · · + |λn|. For various classes of unicyclic graphs, the graphs with maximal energy are determined. Let P 6 n be obtained by connecting a vertex of the circuit C6 with a terminal vertex of the path Pn−6. For n 7, P 6 n has the maximal energy among all connected unic...

Small vertex-transitive directed strongly regular graphs

We consider directed strongly regular graphs de2ned in 1988 by Duval. All such graphs with n vertices, n6 20, having a vertex-transitive automorphism group, are determined with the aid of a computer. As a consequence, we prove the existence of directed strongly regular graphs for three feasible parameter sets listed by Duval. For one parameter set a computer-free proof of the nonexistence is pr...

Finite vertex primitive 2-arc regular graphs

A classification is given of finite graphs that are vertex primitive and 2arc regular. The classification involves various new constructions of interesting 2-arc transitive graphs.