An engineering system may consist of several different types of components, belonging to such physical “domains” as mechanical, electrical, fluid, and thermal. It is termed a multi-domain (or multi-physics) system. The present paper concerns the use of linear graphs (LGs) to generate a minimal model for a multi-physics system. A state-space model has to be a minimal realization. Specifically, the number of state variables in the model should be the minimum number that can completely represent the dynamic state of the system. This choice is not straightforward. Initially, state variables are assigned to all the energy-storage elements of the system. However, some of the energy storage elements may not be independent, and then some of the chosen state variables will be redundant. An approach is presented in the paper, with illustrative examples in the mixed fluid-mechanical domains, to illustrate a way to recognize dependent energy storage elements and thereby obtain a minimal state-space model. System analysis in the frequency domain is known to be more convenient than in the time domain, mainly because the relevant operations are algebraic rather than differential. For achieving this objective, the state space model has to be converted into a transfer function. The direct way is to first convert the state-space model into the input-output differential equation, and then substitute the time derivative by the Laplace variable. This approach is shown in the paper. The same result can be obtained through the transfer function linear graph (TF LG) of the system. In a multi-physics system, first the physical domains have to be converted into an equivalent single domain (preferably, the output domain of the system), when using the method of TFLG. This procedure is illustrated as well, in the present paper.