This volume contains the proceedings of an advanced seminar on graph theory and its applications held in Madison, Wisconsin on October 13-15, 1969 and sponsored by Mathematics Research Center, United States Army, the University of Wisconsin. Ten of the eleven papers presented there are in this volume, documenting one of the most colorful events in recent mathematical memory. These papers provide a survey of the state of the art in graph theory, in hope of advancing mathematical research in the subject.