  In this paper, we study curvature dimension conditions on birth-death
processes which correspond to linear graphs, i.e., weighted graphs supported on
the infinite line or the half line. We give a combinatorial characterization of
Bakry and \'Emery's $CD(K,n)$ condition for linear graphs and prove the
triviality of edge weights for every linear graph supported on the infinite
line $\mathbb{Z}$ with non-negative curvature. Moreover, we show that linear
graphs with curvature decaying not faster than $-R^2$ are stochastically
complete. We deduce a type of Bishop-Gromov comparison theorem for normalized
linear graphs. For normalized linear graphs with non-negative curvature, we
obtain the volume doubling property and the Poincar\'e inequality, which yield
Gaussian heat kernel estimates and parabolic Harnack inequality by Delmotte's
result. As applications, we generalize the volume growth and stochastic
completeness properties to weakly spherically symmetric graphs. Furthermore, we
give examples of infinite graphs with a positive lower curvature bound.
