  This paper presents a comprehensive analysis of the generalized spectral
structure of the modularity matrix $B$, which is introduced by Newman as the
kernel matrix for the quadratic-form expression of the modularity function $Q$
used for community detection. The analysis is then seamlessly extended to the
resolution-parametrized modularity matrix $B(\gamma)$, where $\gamma$ denotes
the resolution parameter. The modularity spectral analysis provides fresh and
profound insights into the $\gamma$-dynamics within the framework of modularity
maximization for community detection. It provides the first algebraic
explanation of the resolution limit at any specific $\gamma$ value. Among the
significant findings and implications, the analysis reveals that (1) the maxima
of the quadratic function with $B(\gamma)$ as the kernel matrix always reside
in the Fiedler space of the normalized graph Laplacian $L$ or the null space of
$L$, or their combination, and (2) the Fiedler value of the graph Laplacian $L$
marks the critical $\gamma$ value in the transition of candidate community
configuration states between graph division and aggregation. Additionally, this
paper introduces and identifies the Fiedler pseudo-set (FPS) as the de facto
critical region for the state transition. This work is expected to have an
immediate and long-term impact on improvements in algorithms for modularity
maximization and on model transformations.
