Another construction of edge-regular graphs with regular cliques

We exhibit a new construction of edge-regular graphs with regular cliques that are not strongly regular. The infinite family of graphs resulting from this construction includes an edge-regular graph with parameters $(24,8,2)$. We also show that edge-regular graphs with $1$-regular cliques that are not strongly regular must have at least $24$ vertices.Comment: 7 page

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

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

Graphs with many valencies and few eigenvalues

Dom de Caen posed the question whether connected graphs with three distinct eigenvalues have at most three distinct valencies. We do not answer this question, but instead construct connected graphs with four and five distinct eigenvalues and arbitrarily many distinct valencies. The graphs with four distinct eigenvalues come from regular two-graphs. As a side result, we characterize the disconnected graphs and the graphs with three distinct eigenvalues in the switching class of a regular two-graph

Oxygen supply and consumption in soilless culture: evaluation of an oxygen simulation model for cucumber

A soil oxygen simulation model (OXSI) was tested and evaluated for evaluating growing media with respect to aeration. In the model, local oxygen concentrations are calculated from coefficients of diffusion and consumption (respiration), assuming equilibrium conditions. Apparent oxygen diffusion coefficients (D) were determined under laboratory conditions in 5 cm high samples at different water contents (-3.2, -10 and -20 cm pressure heads). D values were positively related to air-filled porosity (AFP). For fine-graded perlite D ranged from 9.10-7 at AFP of 34 percent to 5.10-9 m2s-1 at AFP of 19 percent. Possibly due to absence of closed pores in rockwool, the AFP vs. D relation was different for rockwool compared to perlite: D for rockwool ranged from 2.10-6 at AFP of 56 percent to 3.10-9 m2s-1 at AFP of 3 percent. A greenhouse experiment with cucumber was carried out to determine respiration and realised oxygen concentrations. The cucumbers were grown in 20 cm high, 3.5 litre containers filled with fine-graded perlite and supplied with high-frequency irrigation. AFP varied between 25 and 45 percent. At three heights and on four occasions during growth, oxygen concentration ( f volume) in the medium varied between 16.6 and 20 n the perlite. Root respiration of the cucumbers as determined by two independent methods (in vivo and in vitro) ranged from 1.4 to 5.4 10-6 ml.ml-1.s-1. Using these respiration rates, OXSI calculated that no oxygen depletion may occur at D > 1 to 5 10-7 m2s-1, corresponding with an AFP of 30 percent for both perlite and rockwool. Anoxic condtions were calculated for D values of 10-8 m2s-1, corresponding with AFP below 10 percent for rockwool and 20 percent for perlite

Geometric aspects of 2-walk-regular graphs

A $t$-walk-regular graph is a graph for which the number of walks of given length between two vertices depends only on the distance between these two vertices, as long as this distance is at most $t$. Such graphs generalize distance-regular graphs and $t$-arc-transitive graphs. In this paper, we will focus on 1- and in particular 2-walk-regular graphs, and study analogues of certain results that are important for distance regular graphs. We will generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's multiplicity bound and Terwilliger's analysis of the local structure to 2-walk-regular graphs. We will show that 2-walk-regular graphs have a much richer combinatorial structure than 1-walk-regular graphs, for example by proving that there are finitely many non-geometric 2-walk-regular graphs with given smallest eigenvalue and given diameter (a geometric graph is the point graph of a special partial linear space); a result that is analogous to a result on distance-regular graphs. Such a result does not hold for 1-walk-regular graphs, as our construction methods will show