1,439 research outputs found
The Generalized Distance Spectrum of the Join of Graphs
Let G be a simple connected graph. In this paper, we study the spectral properties of the generalized distance matrix of graphs, the convex combination of the symmetric distance matrix D(G) and diagonal matrix of the vertex transmissions Tr(G) . We determine the spectrum of the join of two graphs and of the join of a regular graph with another graph, which is the union of two different regular graphs. Moreover, thanks to the symmetry of the matrices involved, we study the generalized distance spectrum of the graphs obtained by generalization of the join graph operation through their eigenvalues of adjacency matrices and some auxiliary matrices
A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes
We present a version of the weighted cellular matrix-tree theorem that is
suitable for calculating explicit generating functions for spanning trees of
highly structured families of simplicial and cell complexes. We apply the
result to give weighted generalizations of the tree enumeration formulas of
Adin for complete colorful complexes, and of Duval, Klivans and Martin for
skeleta of hypercubes. We investigate the latter further via a logarithmic
generating function for weighted tree enumeration, and derive another
tree-counting formula using the unsigned Euler characteristics of skeleta of a
hypercube and the Crapo -invariant of uniform matroids.Comment: 22 pages, 2 figures. Sections 6 and 7 of previous version simplified
and condensed. Final version to appear in J. Combin. Theory Ser.
A New Large N Expansion for General Matrix-Tensor Models
We define a new large limit for general or
invariant tensor models, based on an enhanced large
scaling of the coupling constants. The resulting large expansion is
organized in terms of a half-integer associated with Feynman graphs that we
call the index. This index has a natural interpretation in terms of the many
matrix models embedded in the tensor model. Our new scaling can be shown to be
optimal for a wide class of non-melonic interactions, which includes all the
maximally single-trace terms. Our construction allows to define a new large
expansion of the sum over diagrams of fixed genus in matrix models with an
additional global symmetry. When the interaction is the
complete vertex of order , we identify in detail the leading order graphs
for a prime number. This slightly surprising condition is equivalent to the
complete interaction being maximally single-trace.Comment: 57 pages, 20 figures (additional discussion in Sec. 4.1.1. and
additional figure (Fig. 5)
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