54 research outputs found
On the Spectrum of Wenger Graphs
Let , where is a prime and is an integer. For ,
let and be two copies of the -dimensional vector spaces over the
finite field . Consider the bipartite graph with partite
sets and defined as follows: a point is adjacent to a line if and only if the
following equalities hold: for . We call the graphs Wenger graphs. In this paper, we determine all
distinct eigenvalues of the adjacency matrix of and their
multiplicities. We also survey results on Wenger graphs.Comment: 9 pages; accepted for publication to J. Combin. Theory, Series
Tur\'an numbers for -free graphs: topological obstructions and algebraic constructions
We show that every hypersurface in contains a large grid,
i.e., the set of the form , with . We use this to
deduce that the known constructions of extremal -free and
-free graphs cannot be generalized to a similar construction of
-free graphs for any . We also give new constructions of
extremal -free graphs for large .Comment: Fixed a small mistake in the application of Proposition
Graphs with few paths of prescribed length between any two vertices
We use a variant of Bukh's random algebraic method to show that for every
natural number there exists a natural number such that, for
every , there is a graph with vertices and edges
with at most paths of length between any two vertices. A result of
Faudree and Simonovits shows that the bound on the number of edges is tight up
to the implied constant.Comment: 8 page
A bound on the number of edges in graphs without an even cycle
We show that, for each fixed , an -vertex graph not containing a cycle
of length has at most edges.Comment: 16 pages, v2 appeared in Comb. Probab. Comp., v3 fixes an error in v2
and explains why the method in the paper cannot improve the power of k
further, v4 fixes the proof of Theorem 12 introduced in v
- …