Title: $Graph Embedding via Topology and Functional Analysis$. Abstract: Graphs have been ubiquitous in Machine Learning due to their versatile nature in modelling real world situations .Graph embedding is an important precursor to using graphs in Machine Learning , and much of performance of algorithms developed later depends heavily on this. However very little theoretical work exists in this area ,  resulting in the proliferation of several benchmarks without any mathematical validation , which is detrimental .In this paper we present an analysis of deterministic graph embedding in general , using tools from Functional Analysis and Topology . We prove several important results pertaining to graph embedding  which may have practical importance .One limitation of our work in it's present form is it's applicable to deterministic embedding approaches only, although we strongly hope to extend it to random graph embedding methods as well in future.We sincerely hope that this work will be beneficial  to researchers working in field of  graph embedding.