The books of [Bapat 2010] or [Godsil and Royle 2013] are excellent sources for more details on graphs and matrices. Spectrum of the graph also plays an important role in studying "zeta function" on graphs (see [Terras 2010]) and uses what is known as the non-backtracking matrix of a graph. As mentioned before, higher-dimensional topological analogues of incidence matrix can be found in the book of [Edelsbrunner 2010]. For a more analytical connection of the Laplacian, see [Grigoryan 2018]. The Laplacian also plays a crucial role in the certain games (Dollar game or abelian sandpile model) defined on graphs and using this one can formulate and prove Riemann-Roch theorem for graphs; see [Corry and Perkinson 2018]. Abelian Sandpile models were originally introduced by physicists to model certain phenomena of ‘particles organizing themselves’ and still many interesting questions remain mathematically unproven about this model. See [Perkinson 2011] for connections between "algebraic geometry" and "sandpiles". See [Bond and Levine 2013] for more general models than abelian sandpiles known as "Abelian networks".
More generally, linear algebra methods have wide usage in combinatorics than those described here via matrices. Two excellent sources are [Babai 2020] and [Jukna 2011].