New arithmetic invariants for cospectral graphs

An invariant for cospectral graphs is a property shared by all cospectral graphs. In this paper, we establish three novel arithmetic invariants for cospectral graphs, revealing deep connections between spectral properties and combinatorial structures. More precisely, one of our main results shows that for any two cospectral graphs $G$ and $H$ with adjacency matrices $A(G)$ and $A(H)$, respectively, the following congruence holds for all integers $m\ge 0$:$$e^{\mathrm{T}}A(G{)}^{m}e\equiv e^{\mathrm{T}}A(H{)}^{m}e(\mathrm{m}\mathrm{o}\mathrm{d}4),$$ where $e$ is the all-one vector. Moreover, we present a number of fascinating applications. Specifically: i) Resolving a conjecture proposed by the third author, we demonstrate that under certain conditions, every graph cospectral with a graph $G$ is determined by its generalized spectrum. ii) We demonstrate that whenever the complements of two trees are cospectral, then one tree has a perfect matching if and only if the other does. An analogous result holds for the existence of triangles in general graphs. iii) An unexpected connection to the polynomial reconstruction problem is also provided, showing that the parity of the constant term of the characteristic polynomial is reconstructible.