On Extremal Unicyclic Molecular Graphs with Prescribed Girth and Minimal Hosoya Index

Ou, Jianping

doi:10.1007/s10910-006-9112-y

Cited by 13 publications

(10 citation statements)

References 12 publications

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“…A particularly interesting additional parameter that does not exist for trees is the girth, i.e., the length of the (unique) cycle. The following theorem (or parts thereof) occurs in several papers: Theorem 8 (See [11,80,82,85,102,103,116]) Among all unicyclic graphs of order n and girth r, the maximum of the Merrifield-Simmons index (which is 2 n−r F r+1 + F r−1 ) and the minimum of the Hosoya index (which is (n − r + 1)F r + 2F r−1 ) are attained by the graph that results from attaching n − r leaves to a single vertex of a cycle C r (see Fig. 8, left).…”

Section: Unicyclic Graphsmentioning

confidence: 99%

“…8, right). Second-and third-smallest or (-largest) and even further values have been determined as well in some of the cases: see [11,80,82,102] and in particular the paper by Ye et al [116], in which the first ≈ r 2 unicyclic graphs of given order n and girth r with respect to the Merrifield-Simmons index (i.e., the graphs with largest, second largest, . .…”

Section: Unicyclic Graphsmentioning

confidence: 99%

“…Theorem 9 (See [11,80,82,85]) Among all unicyclic graphs of order n, the maximum of the Merrifield-Simmons index and the minimum of the Hosoya index (which are 3 · 2 n−3 + 1 and 2n − 2 respectively) are attained for the graph that results from attaching n − 3 leaves to a triangle (the only exception being n = 4, in which case the cycle C 4 also maximizes the…”

Section: Unicyclic Graphsmentioning

confidence: 99%

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Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

2010

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The Hosoya index and the Merrifield-Simmons index are typical examples of graph invariants used in mathematical chemistry for quantifying relevant details of molecular structure. In recent years, quite a lot of work has been done on the extremal problem for these two indices, i.e., the problem of determining the graphs within certain prescribed classes that maximize or minimize the index value. This survey collects and classifies these results, and also provides some useful auxiliary results, tools and techniques that are frequently used in the study of this type of problem.Keywords Hosoya index · Merrifield-Simmons index · Extremal problem · Molecular graphs Mathematics Subject Classification (2000) 92E10 · 05C35 · 05C70 History and Chemical BackgroundIn 1971 the Japanese chemist Haruo Hosoya introduced a molecular-graph based structure descriptor [35], which he named topological index and denoted by Z. He showed that certain physico-chemical properties of alkanes (= saturated hydrocarbons)-in particular, their boiling points-are well correlated with Z. He defined the quantity Z in the following manner.Let G be a (molecular) graph. Denote by m(G, k) the number of ways in which k mutually independent edges can be selected in G. By definition, m(G, 0) = 1 for all graphs, and

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Section: Unicyclic Graphsmentioning

confidence: 99%

Section: Unicyclic Graphsmentioning

confidence: 99%

Section: Unicyclic Graphsmentioning

confidence: 99%

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Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

2010

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“…Extremal graph theory [8] is the study of graphs that are maximal (or minimal) in some way (for example in terms of number of edges) and which satisfy a given property. Extremal graph theory has many applications both in other areas of mathematics and fields including, for example, chemistry [21], biology [22] and cryptography [23].…”

Section: Extremal Graph Problemsmentioning

confidence: 99%

Constraints for symmetry breaking in graph representation

et al. 2018

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Many complex combinatorial problems arising from a range of scientific applications (such as computer networks, mathematical chemistry and bioinformatics) involve searching for an undirected graph satisfying a given property. Since for any possible solution there can be a large number of isomorphic representations, these problems can quickly become intractable. One way to mitigate this problem is to eliminate as many isomorphic copies as possible by breaking symmetry during search-i.e. by introducing constraints that ensure that at least one representative graph is generated for each equivalence class, but not the entire class. The goal is to generate as few members of each class as possible-ideally exactly one: the symmetry break is said to be complete in this case. In this paper we introduce novel, effective and compact, symmetry breaking constraints for undirected graph search. While incomplete, these prove highly beneficial in pruning the search for a graph. We illustrate the application of symmetry breaking in graph representation to resolve several open instances in extremal graph theory. We also illustrate the application of our approach to graph edge coloring problems which exhibit additional symmetries due to the fact that the colors of the edges in any solution can be permuted.

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“…For example, Gutman [6] in 1977 proved that the path is the tree that maximizes the Hosoya index and the star is the tree that minimizes it; while Prodinger and Tichy [16] in 1982 proved that the path minimizes the Merrifield-Simmons index and the star maximizes it among all trees of fixed order. The same pattern also exists in unicyclic graphs and bicyclic graphs, see [1][2][3][4][5]12,13,15,19,20,22]. However, Liu et al [8] in 2015 showed that different patterns appear in tricyclic graphs.…”

Section: Introductionmentioning

confidence: 98%

The Hosoya index and the Merrifield–Simmons index

Huang

Shi

2018

J Math Chem

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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. Sharp lower bounds on the Hosoya index are known for graphs of order n and size m ∈ [n − 1, 2n − 3] ∪ n−1 2 , n 2 ; while sharp upper bounds were only known for graphs of order n and size m ≤ n + 2. We give sharp upper bounds on the Hosoya index for dense graphs with m ≥ n 2 − 2n/3. Moreover, all extreme graphs are also determined.

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On Extremal Unicyclic Molecular Graphs with Prescribed Girth and Minimal Hosoya Index

Cited by 13 publications

References 12 publications

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

Maxima and Minima of the Hosoya Index and the Merrifield-Simmons Index

Constraints for symmetry breaking in graph representation

The Hosoya index and the Merrifield–Simmons index

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