On the extremal Merrifield-Simmons index and Hosoya index of quasi-tree graphs

Authors:
Shuchao Li;Xuechao Li;Wei Jing
Affiliations:
Faculty of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China;Division of Academic Enhancement, The University of Georgia, GA, 30602, USA;Faculty of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, PR China
Venue:
Discrete Applied Mathematics
Year:
2009

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Abstract

It is well known that the two graph invariants, ''the Merrifield-Simmons index'' and ''the Hosoya index'' are important in structural chemistry. A graph G is called a quasi-tree graph, if there exists u"0 in V(G) such that G-u"0 is a tree. In this paper, at first we characterize the n-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Merrifield-Simmons indices. Then we characterize the n-vertex quasi-tree graphs with the largest, the second-largest, the smallest and the second-smallest Hosoya indices, as well as those n-vertex quasi-tree graphs with k pendent vertices having the smallest Hosoya index.