For a graph G with adjacency matrix A(G), the splitting field \(\mathbb {F}(G)\) is defined as the splitting field over \(\mathbb {Q}\) of the characteristic polynomial f(x) of A(G). The extension degree \([\mathbb {F}:\mathbb {Q}]\) is termed the algebraic degree of G and denoted \(\deg (G)\) . A graph G satisfying \(\textrm{deg}(G)=1\) is called an integral graph. Investigating the algebraic degrees of graphs thus generalizes the study of integral graphs. To establish foundational results in this area, for fundamental classes of trees, we derive explicit formulas for \(\textrm{deg}(G)\) and characterize their splitting field. Surprisingly, the algebraic degrees of infinite families of trees are determined by solutions to Pell equations, exposing a profound number-theoretic structure intrinsic to graph spectra. This work provides a new paradigm for classifying graphs via field extensions.
