Unicyclic graphs with given number of pendent vertices and minimal energy

The energy of a graph G, denoted by E(G), is defined to be the sum of absolute values of all eigenvalues of the adjacency matrix of G. Let G(n, l, p) denote the set of all unicyclic graphs on n vertices with girth and pendent vertices being l ( 3) and p ( 1), respectively. More recently, one of the present authors [H. Hua, On minimal energy of unicyclic graphs with prescribed girth and pendent vertices, Match 57 (2007) 351–361] determined the minimal-energy graph in G(n, l, p). In this work, we almost completely solve this problem, cf. Theorem 15. We characterize the graphs having minimal energy among all elements of G(n, p), the set of unicyclic graphs with n vertices and p pendent vertices. Exceptionally, for some values of n and p (see Theorem 15) we reduce the problem to finding the minimal-energy species to only two graphs. © 2007 Elsevier Inc. All rights reserved. AMS classification: 05C35; 05C50; 05C90

[1]  Hongbo Hua,et al.  BIPARTITE UNICYCLIC GRAPHS WITH MAXIMAL, SECOND{MAXIMAL, AND THIRD{MAXIMAL ENERGY , 2007 .

[2]  Ivan Gutman.  Acyclic systems with extremal Hückel π-electron energy , 1977 .

[3]  Yaoping Hou,et al.  Unicyclic graphs with maximal energy , 2002 .

[4]  I. Gutman.  The Energy of a Graph: Old and New Results , 2001 .

[5]  Ivan Gutman,et al.  Topology and stability of conjugated hydrocarbons: The dependence of total π-electron energy on molecular topology , 2005 .

[6]  Bo Zhou,et al.  MINIMAL ENERGY OF BIPARTITE UNICYCLIC GRAPHS OF A GIVEN BIPARTITION , 2022 .

[7]  Hou Yao-ping,et al.  On the spectral radius, k-degree and the upper bound of energy in a graph , 2007 .

[8]  Bolian Liu,et al.  On a Pair of Equienergetic Graphs , 2008 .

[9]  I. Gutman.  Total ?-electron energy of benzenoid hydrohabons , 1983 .

[10]  Hongbo Hua,et al.  Bipartite Unicyclic Graphs with Large Energy , 2007 .

[11]  Ivan Gutman,et al.  Spectra and energies of iterated line graphs of regular graphs , 2005, Appl. Math. Lett..

[12]  I. Gutman,et al.  Mathematical Concepts in Organic Chemistry , 1986 .

[13]  Bo Zhou,et al.  Minimal energy of unicyclic graphs of a given diameter , 2008 .

[14]  Wenshui Lin,et al.  On the extremal energies of trees with a given maximum degree , 2005 .

[15]  Yaoping Hou.  Unicyclic Graphs with Minimal Energy , 2001 .

[16]  Bo Zhou,et al.  On minimal energies of trees of a prescribed diameter , 2006 .

[17]  I. Outman.  Acyclic conjugated molecules, trees and their energies , 1987 .

[18]  B. Devadas Acharya,et al.  EQUIENERGETIC GRAPHS , 2003 .

[19]  Yaoping Hou,et al.  Bicyclic graphs with minimum energy , 2001 .

[20]  F. Tian,et al.  New upper bounds for the energy of graphs , 2004 .

[21]  ANOTHER CLASS OF EQUIENERGETIC GRAPHS , 2004 .