Bounds on the Number of Vertex Independent Sets in a Graph

Pedersen, Anders Degn; Vestergaard, Preben Dahl

doi:10.11650/twjm/1500404576

Cited by 20 publications

(22 citation statements)

References 15 publications

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“…Several authors have studied the number of independent sets in graphs arising from various families (see, for example, [11,12,14,18]). The parameter i(G) is sometimes called the Fibonacci number of a graph [15], since i(P n ) = F (n + 2), where P n is the path with n vertices and F (m) is the mth Fibonacci number. We note in passing that i(G) is the evaluation of the independence polynomial of G at the value 1; see [6,7,8] for more on the independence polynomial.…”

Section: Independence Densitymentioning

confidence: 99%

Independence and chromatic densities of graphs

Bonato

Brown

Kemkes

et al. 2011

Journal of Combinatorics

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We consider graph densities in countably infinite graphs. The independence density of a finite graph G of order n is its proportion of independent sets to all subsets of vertices, while the chromatic density is its proportion of proper n-colourings to all mappings from vertices of G to {1, 2, . . . , n}. For both densities, we extend their definition to countable graphs via limits of chains of finite graphs. We show that independence densities exist for all chains, and are unique regardless of which limiting chain is used. We prove that independence densities are always rational; in fact, we prove the stronger fact that the closure of the set of possible values is contained in the rationals. In contrast, we show that the infinite random graph contains chains realizing all real numbers in [0, 1] as a chromatic density.

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Section: Independence Densitymentioning

confidence: 99%

Independence and chromatic densities of graphs

Bonato

Brown

Kemkes

et al. 2011

Journal of Combinatorics

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“…On the other hand, the minimum for a given stability number is somewhat easier to obtain and given in [86]: apart from the vertices that form the largest independent set, all vertices are mutually connected in the extremal graph. A very similar problem results if one fixes the size of a maximum matching; Yu and Tian [118] consider this in even more generality: in addition to the order n of the graph and its edge-independence number m (i.e., the size of a maximum matching), the cyclomatic number t is fixed (this is equivalent to prescribing the number of edges, which is equal to n + t − 1).…”

Section: General Graphsmentioning

confidence: 99%

“…Finally, a result by Pedersen and Vestergaard [86] deserves to be mentioned: if the girth r and the maximum distance h of a vertex from the unique cycle are fixed, the maximum of the Merrifield-Simmons index is attained for a graph that results from attaching n − r − h single edges as well as a path of length h to one of the vertices of a cycle C r .…”

Section: Merrifield-simmons Index) On the Other Hand The Maximum Ofmentioning

confidence: 99%

“…One such example is a paper by Alameddine [1], in which maximum and minimum of the Merrifield-Simmons index are determined for maximal outerplanar graphs (which are equivalent to triangulations of polygons). Various classes of graphs are studied by Pedersen and Vestergaard in [86], for instance connected bipartite graphs: the maximum and minimum of the Merrifield-Simmons index are obtained for the star and the complete bipartite graph whose two parts are equal in size (or differ at most by one if the total number of vertices is odd). Another remarkable result proven in [86] is the fact that the maximum of the Merrifield-Simmons index for connected claw-free graphs is attained for the path (note that on the other hand, the path minimizes the Merrifield-Simmons index within the class of trees).…”

Section: Other Classes Of 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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“…(Zykov was actually considering cliques in a graph with given clique number, but by complementation this is equivalent to considering independent sets in a graph with given independence number.) Other articles addressing questions of this kind include [9], [11], [14] and [16].Having resolved the question of maximizing i(G) for G in a particular family, it is natural to ask which graph maximizes i t (G), the number of independent sets of size t in G, for each possible t. For many families, it turns out that the graph which maximizes i(G) also maximizes i t (G) for all t. Wingard [17] showed this for trees, Zykov [19] showed this for graphs with a given independence number (see [4] for a short proof), and Cutler and Radcliffe [4] showed this for graphs on a fixed number of edges (again, as a corollary of Kruskal-Katona). In [10], Kahn conjectured that for all 2d|n and all t, no n-vertex, d-regular graph admits more independent sets of size t than the disjoint union of…”

mentioning

confidence: 99%

Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

Engbers

Galvin

2013

Journal of Graph Theory

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Galvin showed that for all fixed δ and sufficiently large n, the n-vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph K δ,n−δ . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples (n, δ, t) with t ≥ 3, no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than K δ,n−δ .Here we make further progress. We show that for all triples (n, δ, t) with δ ≤ 3 and t ≥ 3, no n-vertex graph with minimum degree δ admits more independent sets of size t than K δ,n−δ , and we obtain the same conclusion for δ > 3 and t ≥ 2δ + 1. Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex.

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Bounds on the Number of Vertex Independent Sets in a Graph

Cited by 20 publications

References 15 publications

Independence and chromatic densities of graphs

Independence and chromatic densities of graphs

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

Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

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