Matrix is one of the convenient means for depicting/illustrating a graph on the computer. In this study, the notion graph-theory in conjunction with matrix algebraic operations has been adopted for solving the load-flow problem of three-phase distribution systems (radial and meshed). Five significant matrices, path impedance (PI), loads beyond branch (LB), path drop (PD), slack bus to other buses drop (SBOBD), load flow matrix (LFM), and straight-forward matrix operations have been utilised to attain the load-flow solutions. The aforementioned matrices reveal the system's topology and pertinent information about the operating characteristics of the distribution system during LF studies. This algorithm is formulated entirely on various matrices formulation and computations, even at the stage of upgrading the voltage at every individual bus. Owing to the aforementioned reasons, this LF methodology is computationally efficient for large-sized distribution systems. Moreover, the distributed generations (DGs) modelled as PQ and PV buses are incorporated into the proposed load-flow algorithm. A generalised breakpoint matrix has been derived to compute the mesh breakpoint and PV breakpoint injections simultaneously. The effectiveness of the proposed methodology has been tested on several standard distribution systems. The test outcome shows the viability and accuracy of the proposed method.