We re-visit the asymptotics of a binomial and a Poisson sum which arose as (average) displacement costs when moving randomly placed sensors to anchor positions. The first-order asymptotics of these sums were derived in several stages in a series of recent papers. In this paper, we give a unified approach based on the classical Laplace method with which one can also derive more terms in the asymptotic expansions. Moreover, in a special case, full asymptotic expansions can be given which even hold as identities. This will be proved by a combinatorial approach and systematic ways of computing all coefficients of these identities will be discussed as well.

No abstract

A mixed tree is a tree in which both directed arcs and undirected edges may exist. Let T be a mixed tree with n vertices and m arcs, where an undirected edge is counted twice as arcs. Let A be the adjacency matrix of T . For α ∈ [0, 1], the matrix A α of T is defined to be αD + + (1 − α)A, where D + is the the diagonal out-degree matrix of T . The A α -spectral radius of T is the largest real eigenvalue of A α . We will give a sharp upper bound and a sharp lower bound of the A α -spectral radius of T .

No abstract