2,575 research outputs found
Observability in Connected Strongly Regular Graphs and Distance Regular Graphs
International audienceThis paper concerns the study of observability in consensus networks modeled with strongly regular graphs or distance regular graphs. We first give a Kalman-like simple algebraic criterion for observability in distance regular graphs. This criterion consists in evaluating the rank of a matrix built with the components of the Bose-Mesner algebra associated with the considered graph. Then, we define some bipartite graphs that capture the observability properties of the graph to be studied. In particular, we show that necessary and sufficient observability conditions are given by the nullity of the so-called local bipartite observability graph (resp. local unfolded bipartite observability graph) for strongly regular graphs (resp. distance regular graphs). When the nullity cannot be derived directly from the structure of these bipartite graphs, the rank of the associated bi-adjacency matrix allows evaluating observability. Eventually, as a by-product of the main results we show that non-observability can be stated just by comparing the valency of the graph to be studied with a bound computed from the number of vertices of the graph and its diameter. Similarly nonobservability can also be stated by evaluating the size of the maximum matching in the above mentioned bipartite graphs
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
Geometric aspects of 2-walk-regular graphs
A -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 . Such graphs generalize
distance-regular graphs and -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
An inequality involving the second largest and smallest eigenvalue of a distance-regular graph
For a distance-regular graph with second largest eigenvalue (resp. smallest
eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where
equality only holds when the diameter equals two. Using this inequality we
study distance-regular graphs with fixed second largest eigenvalue.Comment: 15 pages, this is submitted to Linear Algebra and Applications
Shilla distance-regular graphs
A Shilla distance-regular graph G (say with valency k) is a distance-regular
graph with diameter 3 such that its second largest eigenvalue equals to a3. We
will show that a3 divides k for a Shilla distance-regular graph G, and for G we
define b=b(G):=k/a3. In this paper we will show that there are finitely many
Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify
Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will
give a new existence condition for distance-regular graphs, in general.Comment: 14 page
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