<p>Reaction networks are a general framework widely used in modeling diverse phenomena in different science disciplines. The dynamical process of a reaction network endowed with <i>mass-action</i> kinetics is a mass-action system which is an ODE defined by a directed graph, the so-called “reaction graph”. Endotacticity is a graph property used to study persistence and permanence of mass-action systems. In this paper, we provide a detailed characterization of first-order endotactic reaction graphs. Besides, we provide a sufficient condition for endotacticity of reaction networks which are <i>not</i> necessarily of first-order. Such a characterization of a first-order endotactic reaction graph yields the spectral property of the adjacency matrix of the reaction graph. As a consequence, we prove that every first-order <i>endotactic</i> mass-action system as a linear ODE has a weakly reversible deficiency zero realization, and has a unique equilibrium which is exponentially globally asymptotically stable (and is positive) in each (positive) stoichiometric compatibility class. Using a stability result for <i>asymptotically autonomous differential equations</i>, examples are constructed to illustrate that the global stability results can be extended to mass-action systems of higher-order reaction networks modeled by <i>nonlinear</i> ODEs, which are <i>not</i> necessarily endotactic. Different from the classical approaches for proving global asymptotic stability, the proof does not rely on the construction of a Lyapunov function. This paper may serve as a starting point of characterizing higher-order endotactic reaction graphs and studying global stability of mass-action systems in general.</p>