Functional connectivity (FC) between brain regions as manifested via fMRI entails signatures that can be used to identify individuals and decode cognitive tasks. In this work, we use methods from graph structure inference to estimate FC, which is in contrast to the conventional approach of deriving FC via correlation. Furthermore, instead of working on raw (temporal) fMRI data, we infer FC graphs from seed-based co-activation patterns. We also propose a multi-task neural network architecture to jointly perform subject-identification and task-decoding from inferred functional brain graphs. We validate the the developed model on data from 100 subjects from the Human Connectome Project across eight fMRI tasks. Most importantly, our results show the superior task-decoding performance of FC graphs inferred from seed-based activity maps over graphs inferred from raw fMRI data. Furthermore, via gradient-based back-projection, we derive a significance score for inputs to the neural network, and present results showing the differential role of brain connections in subject-identification and task-decoding.