78,303 research outputs found
A Lexicographic product for Signed Graphs
A signed graph is a pair = (G; ), where G = (V (G);E(G)) is a graph
and E(G) {+1;−1} is the sign function on the edges of G. The
notion of composition (also known as lexicographic product) of two signed
graphs and = (H; ) already exists in literature, yet it fails to map
balanced graphs onto balanced graphs. We improve the existing denition
showing that our `new' signature on the lexicographic product of G and
H behaves well with respect to switching equivalence. Signed regularities
and some spectral properties are also discussed
Flow polytopes of signed graphs and the Kostant partition function
We establish the relationship between volumes of flow polytopes associated to
signed graphs and the Kostant partition function. A special case of this
relationship, namely, when the graphs are signless, has been studied in detail
by Baldoni and Vergne using techniques of residues. In contrast with their
approach, we provide entirely combinatorial proofs inspired by the work of
Postnikov and Stanley on flow polytopes. As a fascinating special family of
flow polytopes, we study the Chan-Robbins-Yuen polytopes. Motivated by the
beautiful volume formula for the type version,
where is the th Catalan number, we introduce type and
Chan-Robbins-Yuen polytopes along with intriguing conjectures
pertaining to their properties.Comment: 29 pages, 13 figure
Inside-Out Polytopes
We present a common generalization of counting lattice points in rational
polytopes and the enumeration of proper graph colorings, nowhere-zero flows on
graphs, magic squares and graphs, antimagic squares and graphs, compositions of
an integer whose parts are partially distinct, and generalized latin squares.
Our method is to generalize Ehrhart's theory of lattice-point counting to a
convex polytope dissected by a hyperplane arrangement. We particularly develop
the applications to graph and signed-graph coloring, compositions of an
integer, and antimagic labellings.Comment: 24 pages, 3 figures; to appear in Adv. Mat
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