Both group theory and graph theory are pure mathematics, in the sense that they possess the most elegance and beauty. Unlike some topics in pure math; group theory and graph theory are also very much applied mathematics, most predominant in computer engineering and operations research. Group theory is also fundamental to crystallography and the concept of groups is a foundation to much mathematics. Group theory and graph theory are distinct topics of what we may call discrete mathematics.
A very good intro to group theory for the non-professional is: Groups and Their Graphs, by I Grossman, Magnus W (1975) amazon
Also good: Mathematical Groups, by Tony Barnard et al. amazon
Groups, A Path to Geometry , by R P Burn. amazon
The best intro to graph theory i've read is: Introduction to Graph Theory , by Richard J Trudeau. amazon
Elementary Analysis , by Kenneth A Ross. amazon. This book i read around late 1990s.