7,902 research outputs found
Stability for large forbidden subgraphs
We extend the classical stability theorem of Erdos and Simonovits for
forbidden graphs of logarithmic order.Comment: Some polishing. Updated reference
Revisiting two classical results on graph spectra
Let mu(G) and mu_min(G) be the largest and smallest eigenvalues of the
adjacency matricx of a graph G. We refine quantitatively the following two
results on graph spectra. (i) if H is a proper subgraph of a connected graph G,
then mu(G)>mu(H). (ii) if G is a connected nonbipartite graph, then
mu(G)>-mu_min(G)
Graphs with many copies of a given subgraph
We show that if a graph G of order n contains many copies of a given subgraph
H, then it contains a blow-up of H of order log n
Max k-cut and the smallest eigenvalue
Let be a graph of order and size , and let be the maximum size of a -cut of It is shown that where is the
smallest eigenvalue of the adjacency matrix of
An infinite class of graphs forcing equality in this bound is constructed.Comment: 5 pages. Some typos corrected in v
Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice
We complete the construction of raising and lowering operators, given in a
previous work, for the orthogonal polynomials of hypergeometric type on
non-homogeneous lattice, and extend these operators to the generalized
orthogonal polynomials, namely, those difference of orthogonal polynomials that
satisfy a similar difference equation of hypergeometric type.Comment: LaTeX, 19 pages, (late submission to arXiv.org
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
The trace norm of r-partite graphs and matrices
The trace norm of a graph is the sum of
its singular values, i.e., the absolute values of its eigenvalues. The norm
has been intensively studied under the name
of graph energy, a concept introduced by Gutman in 1978.
This note studies the maximum trace norm of -partite graphs, which raises
some unusual problems for . It is shown that, if is an -partite
graph of order then For some
special this bound is tight: e.g., if is the order of a symmetric
conference matrix, then, for infinitely many there is a graph of
order with Comment: 12 page
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