In this paper, we study a class of matrix functions of the combinatorial Laplacian that preserve its structure, i.e. that define matrices which are positive semidefinite, and which have zero row-sum and non-positive off-diagonal entries. This formulation has the merit of presenting different incarnations of the Laplacian matrix appeared in the recent literature, in a unified framework. For the first time, we apply this family of Laplacian functions to consensus theory, and we show that they leave the agreement value unchanged and offer distinctive advantages in terms of performance and design flexibility. The theory is illustrated via worked examples and numerical experiments featuring four representative Laplacian functions in a shape-based distributed formation control strategy for single-integrator robots.