211,078 research outputs found
Disconnecting strongly regular graphs
In this paper, we show that the minimum number of vertices whose removal
disconnects a connected strongly regular graph into non-singleton components,
equals the size of the neighborhood of an edge for many graphs. These include
blocks graphs of Steiner -designs, many Latin square graphs and strongly
regular graphs whose intersection parameters are at most a quarter of their
valency
Strongly walk-regular graphs
We study a generalization of strongly regular graphs. We call a graph
strongly walk-regular if there is an such that the number of walks of
length from a vertex to another vertex depends only on whether the two
vertices are the same, adjacent, or not adjacent. We will show that a strongly
walk-regular graph must be an empty graph, a complete graph, a strongly regular
graph, a disjoint union of complete bipartite graphs of the same size and
isolated vertices, or a regular graph with four eigenvalues. Graphs from the
first three families in this list are indeed strongly -walk-regular for
all , whereas the graphs from the fourth family are -walk-regular
for every odd . The case of regular graphs with four eigenvalues is the
most interesting (and complicated) one. Such graphs cannot be strongly
-walk-regular for even . We will characterize the case that regular
four-eigenvalue graphs are strongly -walk-regular for every odd ,
in terms of the eigenvalues. There are several examples of infinite families of
such graphs. We will show that every other regular four-eigenvalue graph can be
strongly -walk-regular for at most one . There are several examples
of infinite families of such graphs that are strongly 3-walk-regular. It
however remains open whether there are any graphs that are strongly
-walk-regular for only one particular different from 3
On highly regular strongly regular graphs
In this paper we unify several existing regularity conditions for graphs,
including strong regularity, -isoregularity, and the -vertex condition.
We develop an algebraic composition/decomposition theory of regularity
conditions. Using our theoretical results we show that a family of non rank 3
graphs known to satisfy the -vertex condition fulfills an even stronger
condition, -regularity (the notion is defined in the text). Derived from
this family we obtain a new infinite family of non rank strongly regular
graphs satisfying the -vertex condition. This strengthens and generalizes
previous results by Reichard.Comment: 29 page
Hamiltonian Strongly Regular Graphs
We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected non-Hamiltonian strongly regular graph on fewer than 99 vertices.Distance-regular graphs;Hamilton cycles JEL-code
Strongly Regular Graphs as Laplacian Extremal Graphs
The Laplacian spread of a graph is the difference between the largest
eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the
graph. We find that the class of strongly regular graphs attains the maximum of
largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian
matrices and hence the maximum of Laplacian spreads among all simple connected
graphs of fixed order, minimum degree, maximum degree, minimum size of common
neighbors of two adjacent vertices and minimum size of common neighbors of two
nonadjacent vertices. Some other extremal graphs are also provided.Comment: 11 pages, 4 figures, 1 tabl
Strongly regular edge-transitive graphs
In this paper, we examine the structure of vertex- and edge-transitive
strongly regular graphs, using normal quotient reduction. We show that the
irreducible graphs in this family have quasiprimitive automorphism groups, and
prove (using the Classification of Finite Simple Groups) that no graph in this
family has a holomorphic simple automorphism group. We also find some
constraints on the parameters of the graphs in this family that reduce to
complete graphs.Comment: 23 page
The strongly regular (45,12,3,3) graphs
Using two backtrack algorithms based on dierent techniques, designed and implemented independently, we were able to determine up to isomorphism all strongly regular graphs with parameters v = 45, k = 12, λ = μ = 3. It turns out that there are 78 such graphs, having automorphism groups with sizes ranging from 1 to 51840
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