5 research outputs found
On the extreme eigenvalues of regular graphs
In this paper, we present an elementary proof of a theorem of Serre
concerning the greatest eigenvalues of -regular graphs. We also prove an
analogue of Serre's theorem regarding the least eigenvalues of -regular
graphs: given , there exist a positive constant
and a nonnegative integer such that for any -regular graph
with no odd cycles of length less than , the number of eigenvalues
of such that is at least . This
implies a result of Winnie Li.Comment: accepted to J.Combin.Theory, Series B. added 5 new references, some
comments on the constant c in Section
The chromatic index of strongly regular graphs
We determine (partly by computer search) the chromatic index (edge-chromatic
number) of many strongly regular graphs (SRGs), including the SRGs of degree and their complements, the Latin square graphs and their complements,
and the triangular graphs and their complements. Moreover, using a recent
result of Ferber and Jain it is shown that an SRG of even order , which is
not the block graph of a Steiner 2-design or its complement, has chromatic
index , when is big enough. Except for the Petersen graph, all
investigated connected SRGs of even order have chromatic index equal to their
degree, i.e., they are class 1, and we conjecture that this is the case for all
connected SRGs of even order.Comment: 10 page
On a conjecture of Brouwer involving the connectivity of strongly regular graphs
In this paper, we study a conjecture of Andries E. Brouwer from 1996
regarding the minimum number of vertices of a strongly regular graph whose
removal disconnects the graph into non-singleton components.
We show that strongly regular graphs constructed from copolar spaces and from
the more general spaces called -spaces are counterexamples to Brouwer's
Conjecture. Using J.I. Hall's characterization of finite reduced copolar
spaces, we find that the triangular graphs , the symplectic graphs
over the field (for any prime power), and the
strongly regular graphs constructed from the hyperbolic quadrics
and from the elliptic quadrics over the field ,
respectively, are counterexamples to Brouwer's Conjecture. For each of these
graphs, we determine precisely the minimum number of vertices whose removal
disconnects the graph into non-singleton components. While we are not aware of
an analogue of Hall's characterization theorem for -spaces, we show
that complements of the point graphs of certain finite generalized quadrangles
are point graphs of -spaces and thus, yield other counterexamples to
Brouwer's Conjecture.
We prove that Brouwer's Conjecture is true for many families of strongly
regular graphs including the conference graphs, the generalized quadrangles
graphs, the lattice graphs, the Latin square graphs, the strongly
regular graphs with smallest eigenvalue -2 (except the triangular graphs) and
the primitive strongly regular graphs with at most 30 vertices except for few
cases.
We leave as an open problem determining the best general lower bound for the
minimum size of a disconnecting set of vertices of a strongly regular graph,
whose removal disconnects the graph into non-singleton components.Comment: 25 pages, 1 table; accepted to JCTA; revised version contains a new
section on copolar and Delta space
The graphs with all but two eigenvalues equal to or
We determine all graphs for which the adjacency matrix has at most two
eigenvalues (multiplicities included) not equal to , or , and determine
which of these graphs are determined by their adjacency spectrum