a b s t r a c tAn L(2, 1)-labeling of a graph G is a function f from the vertex set of G to the set of where d(x, y) denotes the distance between the pair of vertices x, y. The lambda number of G, denoted λ(G), is the minimum range of labels used over all L(2,1)-labelings of G. An L(2,1)-labeling of G which achieves the range λ(G) is referred to as a λ-labeling. A hole of an L(2,1)-labeling is an unused integer within the range of integers used. The hole index of G, denoted ρ(G), is the minimum number of holes taken over all its λ-labelings. An island of a given λ-labeling of G with ρ(G) holes is a maximal set of consecutive integers used by the labeling. Georges and Mauro [J.P. Georges, D.W. Mauro, On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] inquired about the existence of a connected graph G with ρ(G) ≥ 1 possessing two λ-labelings with different ordered sequences of island cardinalities. This paper provides an infinite family of such graphs together with their lambda numbers and hole indices. Key to our discussion is the determination of the path covering number of certain 2-sparse graphs, that is, graphs containing no pair of adjacent vertices of degree greater than 2.

We simulate a simple nanocomposite consisting of a single spherical nanoparticle surrounded by coarse-grained polymer chains that are composed of two monomer types differing only in their interactions with the nanoparticle. We measure the atomic stress fluctuations and use them to estimate the local stress autocorrelation as a function of distance from the nanoparticle. This local stress autocorrelation is substituted into the well-known relationship between the bulk stress autocorrelation and the bulk (frequency-dependent) dynamic modulus, and the result is treated as an estimate of the local dynamic modulus. This allows us to examine the effect of adjusting copolymer sequence on estimations of local storage and loss modulus as a function of distance from the nanoparticle. Notably, we find certain blocky copolymer sequences can lead to a higher tan(δ) (hysteresis) in the interphase than either homopolymer system, suggesting that tuning the copolymer sequence could allow for significant control over nanocomposite dynamics.

We simulate a simple nanocomposite consisting of a single spherical nanoparticle surrounded by coarse-grained polymer chains. The polymers are composed of two different monomer types that differ only in their interaction strengths with the nanoparticle. We examine the effect of adjusting copolymer sequence on the structure as well as the end-to-end vector autocorrelation, bond vector autocorrelation, and self-intermediate scattering function relaxation times as a function of distance from the nanoparticle surface. We show how the range and magnitude of the interphase of slowed dynamics surrounding the nanoparticle depend strongly on sequence blockiness. We find that, depending on block length, blocky copolymers can have faster or slower dynamics than a random copolymer. Certain blocky copolymer sequences lead to relaxation times near the nanoparticle surface that are slower than those of either homopolymer system. Thus, tuning copolymer sequence could allow for significant control over the nanocomposite behavior.