Hosoya index of unicyclic graphs with prescribed pendent vertices

2007 ◽  
Vol 43 (2) ◽  
pp. 831-844 ◽  
Author(s):  
Hongbo Hua
Keyword(s):  
Hosoya Index ◽  
Unicyclic Graphs ◽  
Pendent Vertices

scholarly journals Note on unicyclic graphs with given number of pendent vertices and minimal energy

2010 ◽  
Vol 433 (7) ◽  
pp. 1381-1387 ◽  
Author(s):  
Bofeng Huo ◽  
Shengjin Ji ◽  
Xueliang Li
Keyword(s):  
Minimal Energy ◽  
Unicyclic Graphs ◽  
Pendent Vertices

scholarly journals On the Laplacian Coefficients and Laplacian-Like Energy of Unicyclic Graphs withnVertices andmPendent Vertices

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Xinying Pai ◽  
Sanyang Liu
Keyword(s):  
Characteristic Polynomial ◽  
Laplacian Matrix ◽  
Unicyclic Graphs ◽  
Pendent Vertices

LetΦ(G,λ)=det(λIn-L(G))=∑k=0n(-1)kck(G)λn-kbe the characteristic polynomial of the Laplacian matrix of a graphGof ordern. In this paper, we give four transforms on graphs that decrease all Laplacian coefficientsck(G)and investigate a conjecture A. Ilić and M. Ilić (2009) about the Laplacian coefficients of unicyclic graphs withnvertices andmpendent vertices. Finally, we determine the graph with the smallest Laplacian-like energy among all the unicyclic graphs withnvertices andmpendent vertices.

scholarly journals The Maximum Hosoya Index of Unicyclic Graphs with Diameter at Most Four

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1034 ◽  
Author(s):  
Weijun Liu ◽  
Jingwen Ban ◽  
Lihua Feng ◽  
Tao Cheng ◽  
Frank Emmert-Streib ◽  
...  
Keyword(s):  
Small Diameter ◽  
Hosoya Index ◽  
Unicyclic Graphs

The Hosoya index of a graph is defined by the total number of the matchings of the graph. In this paper, we determine the maximum Hosoya index of unicyclic graphs with n vertices and diameter 3 or 4. Our results somewhat answer a question proposed by Wagner and Gutman in 2010 for unicyclic graphs with small diameter.

scholarly journals Erratum: Liu, W.; Ban, J.; Feng, L.; Cheng, T.; Emmert-Streib, F.; Dehmer, M. The Maximum Hosoya Index of Unicyclic Graphs with Diameter at Most Four. Symmetry 2019, 11, 1034

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1496
Author(s):  
Weijun Liu ◽  
Jingwen Ban ◽  
Lihua Feng ◽  
Tao Cheng ◽  
Frank Emmert-Streib ◽  
...  
Keyword(s):  
Hosoya Index ◽  
Unicyclic Graphs

The authors wish to make the following corrections to their paper [...]

scholarly journals The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs

2019 ◽  
Vol 35 ◽  
pp. 449-467
Author(s):  
Zhibin Du
Keyword(s):  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian ◽  
Pendent Vertices ◽  
Eigenvalues Of Graphs

Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.

scholarly journals The largest n - 1 Hosoya indices of unicyclic graphs

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2573-2581 ◽  
Author(s):  
Guihai Yu ◽  
Lihua Feng ◽  
Aleksandar Ilic
Keyword(s):  
Independent Sets ◽  
Hosoya Index ◽  
Unicyclic Graphs ◽  
Molecular Graphs ◽  
Appl Math

The Hosoya index Z(G) of a graph G is defined as the total number of edge independent sets of G. In this paper, we extend the research of [J. Ou, On extremal unicyclic molecular graphs with maximal Hosoya index, Discrete Appl. Math. 157 (2009) 391-397.] and [Y. Ye, X. Pan, H. Liu, Ordering unicyclic graphs with respect to Hosoya indices and Merrifield-Simmons indices, MATCH Commun. Math. Comput. Chem. 59 (2008) 191-202.] and order the largest n - 1 unicyclic graphs with respect to the Hosoya index.

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