533 research outputs found
Square Property, Equitable Partitions, and Product-like Graphs
Equivalence relations on the edge set of a graph that satisfy restrictive
conditions on chordless squares play a crucial role in the theory of Cartesian
graph products and graph bundles. We show here that such relations in a natural
way induce equitable partitions on the vertex set of , which in turn give
rise to quotient graphs that can have a rich product structure even if
itself is prime.Comment: 20 pages, 6 figure
ON VULNERABILITY MEASURES OF NETWORKS
As links and nodes of interconnection networks are exposed to failures, one of the most important features of a practical networks design is fault tolerance. Vulnerability measures of communication
networks are discussed including the connectivities, fault diameters, and measures based on Hosoya-Wiener polynomial. An upper bound for the edge fault diameter of product graphs is proved
ON VULNERABILITY MEASURES OF NETWORKS
As links and nodes of interconnection networks are exposed to failures, one of the most important features of a practical networks design is fault tolerance. Vulnerability measures of communication
networks are discussed including the connectivities, fault diameters, and measures based on Hosoya-Wiener polynomial. An upper bound for the edge fault diameter of product graphs is proved
The edge fault-diameter of Cartesian graph bundles
AbstractA Cartesian graph bundle is a generalization of a graph covering and a Cartesian graph product. Let G be a kG-edge connected graph and D̄c(G) be the largest diameter of subgraphs of G obtained by deleting c<kG edges. We prove that D̄a+b+1(G)≤D̄a(F)+D̄b(B)+1 if G is a graph bundle with fibre F over base B, a<kF, and b<kB. As an auxiliary result we prove that the edge-connectivity of graph bundle G is at least kF+kB
Weil-Petersson perspectives
We highlight recent progresses in the study of the Weil-Petersson (WP)
geometry of finite dimensional Teichm\"{u}ller spaces. For recent progress on
and the understanding of infinite dimensional Teichm\"{u}ller spaces the reader
is directed to the recent work of Teo-Takhtajan. As part of the highlight, we
also present possible directions for future investigations.Comment: 18 page
A diagrammatic view of differential equations in physics
Presenting systems of differential equations in the form of diagrams has
become common in certain parts of physics, especially electromagnetism and
computational physics. In this work, we aim to put such use of diagrams on a
firm mathematical footing, while also systematizing a broadly applicable
framework to reason formally about systems of equations and their solutions.
Our main mathematical tools are category-theoretic diagrams, which are well
known, and morphisms between diagrams, which have been less appreciated. As an
application of the diagrammatic framework, we show how complex, multiphysical
systems can be modularly constructed from basic physical principles. A wealth
of examples, drawn from electromagnetism, transport phenomena, fluid mechanics,
and other fields, is included.Comment: 69 page
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