87 research outputs found
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
The matching polynomial of a regular graph
AbstractThe matching polynomial of a graph has coefficients that give the number of matchings in the graph. For a regular graph, we show it is possible to recover the order, degree, girth and number of minimal cycles from the matching polynomial. If a graph is characterized by its matching polynomial, then it is called matching unique. Here we establish the matching uniqueness of many specific regular graphs; each of these graphs is either a cage, or a graph whose components are isomorphic to Moore graphs. Our main tool in establishing the matching uniqueness of these graphs is the ability to count certain subgraphs of a regular graph
A remark on partial linear spaces of girth with an application to strongly regular graphs
We derive a lower bound on the number of points of a partial linear space of girth 5. As an application, certain strongly regular graphs with=2 are ruled out by observing that the first subconstituents are partial linear spaces
The distance-regular graphs such that all of its second largest local eigenvalues are at most one
In this paper, we classify distance regular graphs such that all of its
second largest local eigenvalues are at most one. Also we discuss the
consequences for the smallest eigenvalue of a distance-regular graph. These
extend a result by the first author, who classified the distance-regular graph
with smallest eigenvalue .Comment: 16 pages, this is submitted to Linear Algebra and Application
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