Graph Signal Processing (GSP) and Graph Neural Networks (GNNs) rely fundamentally on the matrix representation of the underlying graph topology. This representation defines key operators such as the graph Fourier transform, spectral filtering, and convolution. Existing parameterized operator families interpolate only partial subsets of classical graph matrices, while broader formulations become non-compact when representing transition-type operators, limiting both theoretical analysis and stable learning. To address this issue, we propose the Generalized Linear Graph Representation (GLGR), denoted by $\mathbf{Q}_{α,l}$, as a compact two-parameter operator family defined on a bounded linear domain. GLGR unifies major classical operators together with transition-type operators without requiring asymptotic parameters. Theoretically, we show that $\mathbf{Q}_{α,l}$ admits a variational decomposition balancing local smoothness and global degree-weighted energy, derive spectral perturbation bounds, and establish graph-aware sufficient conditions for positive semi-definiteness. Building on this formulation, we develop Adaptive GLGR Convolution (AG-Conv), which makes the propagation operator itself learnable within end-to-end GNNs. Experiments on graph classification and node classification benchmarks show that GLGR improves both fixed-operator representation search and adaptive graph learning across multiple backbones.