  In this paper, we propose a class of graphs $G^{\star}(m,t)$ and first study
some structural properties, such as, average degree, on them. The results show
that (1) graphs $G^{\star}(m,t)$ have density feature because of their average
degrees proportional to time step $t$ not to a constant in the large graph size
limit, (2) graphs $G^{\star}(m,t)$ obey the power-law distribution with
exponent equal to $2$, which is rarely found in most previous scale-free
models, (3) graphs $G^{\star}(m,t)$ display small-world property in terms of
ultra-small diameter and higher clustering coefficient, and (4) graphs
$G^{\star}(m,t)$ possess disassortative structure with respect to Pearson
correlation coefficient smaller than zero. In addition, we consider the
trapping problem on the proposed graphs $G^{\star}(m,t)$ and then find that
they all have more optimal trapping efficiency by means of their own average
trapping time achieving the theoretical lower bound, a phenomenon that is
seldom observed in existing scale-free models. We conduct extensive simulations
that are consistent with our theoretical analysis.
