98 research outputs found
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
A Kruskal–Katona type theorem for graphs
AbstractA bound on consecutive clique numbers of graphs is established. This bound is evaluated and shown to often be much better than the bound of the Kruskal–Katona theorem. A bound on non-consecutive clique numbers is also proven
Dirac-type theorems in random hypergraphs
For positive integers and divisible by , let be the
minimum -degree ensuring the existence of a perfect matching in a
-uniform hypergraph. In the graph case (where ), a classical theorem of
Dirac says that . However, in general, our
understanding of the values of is still very limited, and it is an
active topic of research to determine or approximate these values. In this
paper we prove a "transference" theorem for Dirac-type results relative to
random hypergraphs. Specifically, for any and any
"not too small" , we prove that a random -uniform hypergraph with
vertices and edge probability typically has the property that every
spanning subgraph of with minimum degree at least
has a perfect matching. One interesting aspect of
our proof is a "non-constructive" application of the absorbing method, which
allows us to prove a bound in terms of without actually knowing
its value
Identifying long cycles in finite alternating and symmetric groups acting on subsets
Let be a permutation group on a set , which is permutationally
isomorphic to a finite alternating or symmetric group or acting on
the -element subsets of points from , for some arbitrary but
fixed . Suppose moreover that no isomorphism with this action is known. We
show that key elements of needed to construct such an isomorphism
, such as those whose image under is an -cycle or
-cycle, can be recognised with high probability by the lengths of just
four of their cycles in .Comment: 45 page
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