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Tricyclic graphs with maximum Merrifield–Simmons index

Zhongxun Zhu, Shuchao Li, Liansheng Tan
2010 Discrete Applied Mathematics  
., ordered the unicyclic graphs with given girth according to the Merrifield-Simmons index in [24] .  ...  In [17] , Li et al. characterized the tree with the maximal Merrifield-Simmons index among the trees with given diameter.  ... 
doi:10.1016/j.dam.2009.09.001 fatcat:zrglxwy3vvfwjpwkvnvckac72u


2010 Discrete Applied Mathematics  
Tan Tricyclic graphs with maximum Merrifield-Simmons index 204 Notes M. Atapour, S.M. Sheikholeslami, A.N. Ghameshlou and L. Volkmann Signed star domatic number of a graph 213 A. Behtoei, M.  ...  Taeri A characterization of block graphs 219 M. Fliedner, N. Boysen and A. Scholl Solving symmetric mixed-model multi-level just-in-time scheduling problems 222 H.  ... 
doi:10.1016/s0166-218x(09)00463-6 fatcat:hfdwh36urzabrjwlzeueqbvwsa

The Hosoya index and the Merrifield–Simmons index

Yufeng Huang, Lingsheng Shi, Xingyu Xu
2018 Journal of Mathematical Chemistry  
In this article, we give sharp bounds on the Hosoya index and the Merrifield-Simmons index for connected graphs of fixed size.  ...  As a consequence, we determine all connected graphs of any fixed order and size which maximize the Merrifield-Simmons index.  ...  to the graph with maximum Merrifield-Simmons index.  ... 
doi:10.1007/s10910-018-0937-y fatcat:2zbuo7vqu5b6bmp2356usch6vi

The number of independent sets in a connected graph and its complement

Yarong Wei, Yumei Hu
2018 The Art of Discrete and Applied Mathematics  
Next, we obtain the minimum and maximum value of i(G)+i(G), where graph G is a tree T with connected complement and a unicyclic graph G with connected complement, respectively.  ...  In each case, we characterize the extremal graphs. Finally, we establish an upper bound on the i(G) in terms of the Wiener polarity index.  ...  Then we obtain i(G) + i(G) ≥ 2n + 2D(G) + 2D(G) − 4 with equality if and only if G ∼ = P 4 . Also, we conjecture that the extremal graph which reaches the maximum value of i(G) + i(G) is S 2,n−2 .  ... 
doi:10.26493/2590-9770.1258.c2b fatcat:byqtsmw4rbc3la63vgbllko4bi

Extremal Graph Theory for Degree Sequences [article]

Xiao-Dong Zhang
2015 arXiv   pre-print
eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences.  ...  In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet  ...  Andriantiana [1] proved that the greedy tree has minimum energy, Hosoya index and maximum Merrifield-Simmons index. Theorem 4.11 [1] Let π be a tree degree sequence.  ... 
arXiv:1510.01903v1 fatcat:j7dpgr62djf3vl377nppnbn3na

On the number of independent sets in cycle-separated tricyclic graphs

Ardeshir Dolati
2011 Computers and Mathematics with Applications  
A cycle-separated tricyclic graph (CSTC graph) is a connected simple graph with n vertices and n + 2 edges whose subgraph induced by its cycles consists of three disjoint cycles.  ...  We show that the tight upper bound for the number of independent sets in the n-vertex CSTC graphs is 48 × 2 n−9 + 9 (for n ≥ 9); we also characterize the extremal graph with respect to the aforementioned  ...  We denote the number of independent sets in a graph G by i(G). This is called the Fibonacci number or Merrifield-Simmons index of G, too.  ... 
doi:10.1016/j.camwa.2011.01.021 fatcat:sgfdj7pmmbdr3gb3sodtg55zly

Trees with Given Stability Number and Minimum Number of Stable Sets

Véronique Bruyère, Gwenaël Joret, Hadrien Mélot
2011 Graphs and Combinatorics  
The invariant F (G) is also known as the Merrifield-Simmons index or σ-index of the graph G [14] .  ...  Other classes of graphs have been considered as well; this includes unicyclic graphs [10, 15, 16, 19] , bicyclic graphs [3] , tricyclic graphs [22] , quasi-trees [9] , maximal outerplanar graphs [  ... 
doi:10.1007/s00373-011-1041-2 fatcat:g3z4dlmakfdghcllkeubeuu4ei

The tight upper bound for the number of matchings of tricyclic graphs

Ardeshir Dolati, Somayyeh Golalizadeh
2012 unpublished
We also characterize the n-vertex simple connected tricyclic graph for which the bound is best possible.  ...  In this paper, we determine the tight upper bound for the number of matchings of connected n-vertex tricyclic graphs.  ...  tricyclic graph with the largest z-index.  ...