Book

This monograph develops a systematic operator-theoretic framework for fractional differential equations posed on quantum graphs — metric graphs equipped with differential operators acting edgewise and coupled through vertex conditions. Building on the classical theory of quantum graphs and spectral graph theory, the book extends the field in five directions: a discrete quantum graph Laplacian built from Hermitian, phase-carrying edge weights; its fractional spectral power; a non-singular-kernel Caputo–Fabrizio analogue; a Mittag-Leffler-kernel Atangana–Baleanu analogue; and a variable-order operator whose fractional index is itself a function on the vertex set. A unifying operator is proposed that reduces to each of these constructions under specific parameter limits. Each operator is examined along the same six axes — definition, functional-analytic setting, spectral structure, well-posedness, stability, and numerical discretization — with a worked six-vertex pharmacological interaction network carried throughout as a running illustrative example. Intended for readers with a first graduate course in functional analysis, the book develops fractional calculus and quantum graph prerequisites from first principles.