260 research outputs found
A result on polynomials derived via graph theory
We present an example of a result in graph theory that is used to obtain a
result in another branch of mathematics. More precisely, we show that the
isomorphism of certain directed graphs implies that some trinomials over finite
fields have the same number of roots
Merging the A- and Q-spectral theories
Let be a graph with adjacency matrix , and let
be the diagonal matrix of the degrees of The signless
Laplacian of is defined as .
Cvetkovi\'{c} called the study of the adjacency matrix the %
\textit{-spectral theory}, and the study of the signless Laplacian--the
\textit{-spectral theory}. During the years many similarities and
differences between these two theories have been established. To track the
gradual change of into in this paper it
is suggested to study the convex linear combinations of and defined by This study sheds new light
on and , and yields some surprises, in
particular, a novel spectral Tur\'{a}n theorem. A number of challenging open
problems are discussed.Comment: 26 page
Upper bound theorem for odd-dimensional flag triangulations of manifolds
We prove that among all flag triangulations of manifolds of odd dimension
2r-1 with sufficiently many vertices the unique maximizer of the entries of the
f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses
methods from extremal graph theory.Comment: Clarifications and new references, title has change
Phase transitions in the Ramsey-Turán theory
Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdos and Sós asked if RT (n, K5, c√
n) = o (n2) for some constant c. We answer this question by proving the stronger RT(n, K5, o (√n log n)) = o(n2). It is known that RT (n, K5, c√n log n
)= n2/4 + o (n2) for c > 1, so one can say that K5 has a Ramsey-Turán-phase transition at c√n log n. We extend this result to several other Kp's and functions f(n), determining many more phase transitions. We shall formulate
several open problems, in particular, whether variants of the Bollobás-Erdos graph, which is a geometric construction, exist to give good lower bounds
on RT (n, Kp, f(n)) for various pairs of p and f(n). These problems are studied in depth by Balogh-HuSimonovits, where among others, the Szemerédi's Regularity Lemma and the Hypergraph Dependent
Random Choice Lemma are used.National Science Foundatio
The codegree threshold of
The codegree threshold of a -graph is the
minimum such that every -graph on vertices in which every pair
of vertices is contained in at least edges contains a copy of as a
subgraph. We study when , the -graph on
vertices with edges. Using flag algebra techniques, we prove that if is
sufficiently large then .
This settles in the affirmative a conjecture of Nagle from 1999. In addition,
we obtain a stability result: for every near-extremal configuration , there
is a quasirandom tournament on the same vertex set such that is close
in the edit distance to the -graph whose edges are the cyclically
oriented triangles from . For infinitely many values of , we are further
able to determine exactly and to show that
tournament-based constructions are extremal for those values of .Comment: 31 pages, 7 figures. Ancillary files to the submission contain the
information needed to verify the flag algebra computation in Lemma 2.8.
Expands on the 2017 conference paper of the same name by the same authors
(Electronic Notes in Discrete Mathematics, Volume 61, pages 407-413
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