310 research outputs found
Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs
AbstractLet Γ be a d-bounded distance-regular graph with diameter d⩾3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L′(x,i) by inclusion or reverse inclusion, L′(x,i) is denoted by LO′(x,i) or LR′(x,i). We prove that LO′(x,i) and LR′(x,i) are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of LO′(x,i)
Random walks on graphs: ideas, techniques and results
Random walks on graphs are widely used in all sciences to describe a great
variety of phenomena where dynamical random processes are affected by topology.
In recent years, relevant mathematical results have been obtained in this
field, and new ideas have been introduced, which can be fruitfully extended to
different areas and disciplines. Here we aim at giving a brief but
comprehensive perspective of these progresses, with a particular emphasis on
physical aspects.Comment: LateX file, 34 pages, 13 jpeg figures, Topical Revie
A trace on fractal graphs and the Ihara zeta function
Starting with Ihara's work in 1968, there has been a growing interest in the
study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and
Terras, Mizuno and Sato, to name just a few authors. Then, Clair and
Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by
a discrete group of automorphisms. The main formula in all these treatments
establishes a connection between the zeta function, originally defined as an
infinite product, and the Laplacian of the graph. In this article, we consider
a different class of infinite graphs. They are fractal graphs, i.e. they enjoy
a self-similarity property. We define a zeta function for these graphs and,
using the machinery of operator algebras, we prove a determinant formula, which
relates the zeta function with the Laplacian of the graph. We also prove
functional equations, and a formula which allows approximation of the zeta
function by the zeta functions of finite subgraphs.Comment: 30 pages, 5 figures. v3: minor corrections, to appear on Transactions
AM
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