For δ ≥ 1 and n ≥ 1, we examine the simplicial complex of graphs on n vertices in which each vertex has degree at most δ; we identify a given graph with its edge set and admit one loop at each vertex. δ = 1 yields the matching complex, and it is known that there is 3-torsion in degree d of the homology of this complex whenever n−4 3 ≤ d ≤ n−6 2 . We establish similar bounds for δ ≥ 2. Specifica...

Motivated by Khovanov homology and relations between the Jones polynomial and graph polynomials, we construct a homology theory for embedded graphs from which the chromatic polynomial can be recovered as the Euler characteristic. For plane graphs, we show that our chromatic homology can be recovered from the Khovanov homology of an associated link. We apply this connection with Khovanov homolog...

We survey results and techniques in the topological study of simplicial complexes of (di-, multi-, hyper-)graphs whose node degrees are bounded from above. These complexes have arisen is a variety of contexts in the literature. The most wellknown examples are the matching complex and the chessboard complex. The topics covered here include computation of Betti numbers, representations of the sym...

We make two observations about the zeta function of a graph. First we show how Bass’s proof of Ihara’s formula fits into the framework of torsion of complexes. Second, we show how in the special case of those graphs that are quotients of the Bruhat-Tits tree for SL(2, K) for a local nonarchimedean field K, the zeta function has a natural expression in terms of the L-functions of Coexter systems.

A graphs of rank n (homotopy equivalent to a wedge of n circles) without “separating edges” has a canonical n-dimensional compact C manifold thickening. This implies that the canonical homomorphism φ : Out(Fn) → GL(n,Z) is trivial in rational cohomology in the stable range answering a question raised in [6]. Another consequence of the construction is the existence of higher Reidemeister torsion...

A finite simple graph ? determines a quotient P? of the pure braid group, called graphic arrangement group. We analyze homomorphisms these groups defined by deletion sets vertices, using methods developed in prior joint work with R. Randell. show that, for K4-free ?, product maps is injective, embedding free groups. Then residually free, torsion-free, torsion-free nilpotent, and acts properly o...

let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...

Background: Left ventricular torsion is one of the most important biomechanical parameters of heart that routinely is measured in short axis view.  A review of the literature has indicated that assessment of left ventricular torsion in short axis view has some limitations. In the present study, we evaluated whether torsion angle assessment in long axis view can be used as a diagnostic biomechan...

Let S be a finite generating set of a torsion-free, nilpotent group G. We show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is, every automorphism of the graph is obtained by composing a group automorphism with multiplication by an element of the group.) More generally, we show that if Cay(G1;S1) and Cay(G2;S2) are connected Cayley graphs of finite valency on two nilpot...