1,088 research outputs found
Equiangular lines in Euclidean spaces
We obtain several new results contributing to the theory of real equiangular
line systems. Among other things, we present a new general lower bound on the
maximum number of equiangular lines in d dimensional Euclidean space; we
describe the two-graphs on 12 vertices; and we investigate Seidel matrices with
exactly three distinct eigenvalues. As a result, we improve on two
long-standing upper bounds regarding the maximum number of equiangular lines in
dimensions d=14, and d=16. Additionally, we prove the nonexistence of certain
regular graphs with four eigenvalues, and correct some tables from the
literature.Comment: 24 pages, to appear in JCTA. Corrected an entry in Table
Characterizing Block Graphs in Terms of their Vertex-Induced Partitions
Given a finite connected simple graph with vertex set and edge
set , we will show that
the (necessarily unique) smallest block graph with vertex set whose
edge set contains is uniquely determined by the -indexed family of the various partitions
of the set into the set of connected components of the
graph ,
the edge set of this block graph coincides with set of all -subsets
of for which and are, for all , contained
in the same connected component of ,
and an arbitrary -indexed family of
partitions of the set is of the form for some
connected simple graph with vertex set as above if and only if,
for any two distinct elements , the union of the set in
that contains and the set in that contains coincides with
the set , and holds for all .
As well as being of inherent interest to the theory of block graphs, these
facts are also useful in the analysis of compatible decompositions and block
realizations of finite metric spaces
Characterizing block graphs in terms of their vertex-induced partitions
Block graphs are a generalization of trees that arise in areas such as metric graph theory, molecular graphs, and phylogenetics. Given a finite connected simple graph with vertex set and edge set , we will show that the (necessarily unique) smallest block graph with vertex set whose edge set contains is uniquely determined by the -indexed family \Pp_G =\big(\pi_v)_{v \in V} of the partitions of the set into the set of connected components of the graph . Moreover, we show that an arbitrary -indexed family \Pp=(\p_v)_{v \in V} of partitions \p_v of the set is of the form \Pp=\Pp_G for some connected simple graph with vertex set as above if and only if, for any two distinct elements , the union of the set in \p_v that contains and the set in \p_u that contains coincides with the set , and \{v\}\in \p_v holds for all . As well as being of inherent interest to the theory of block graphs,these facts are also useful in the analysis of compatible decompositions of finite metric spaces
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