Graphs with the second largest number of maximal independent sets

Let G be a simple undirected graph. Denote by (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. Erd?s and Moser raised the problem of determining the maximum value of among all graphs of order n and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of graphs, including trees, forests, bipartite graphs, connected graphs, (connected) triangle-free graphs, (connected) graphs with at most one cycle, and recently, (connected) graphs with at most r cycles. In this paper we determine the second largest value of and xi(G) among all graphs of order n. Moreover, the extremal graphs achieving these values are also determined.     

Trees with the second largest number of maximal independent sets

A maximal independent set is an independent set that is not a proper subset of any other independent set. In this paper, we determine the second largest number of maximal independent sets among all trees and forests of order n≥4. We also characterize those extremal graphs achieving these values.     

The number of independent sets in unicyclic graphs with a given diameter

Let U_n,d denote the set of unicyclic graphs with a given diameter d. In this paper, the unique unicyclic graph in U_n,d with the maximum number of independent sets, is characterized.     

Polar graphs and maximal independent sets

We study two central problems of algorithmic graph theory: finding maximum and minimum maximal independent sets. Both problems are known to be NP-hard in general. Moreover, they remain NP-hard in many special classes of graphs. For instance, the problem of finding minimum maximal independent sets has been recently proven to be NP-hard in the class of so-called (1,2)-polar graphs. On the other hand, both problems can be solved in polynomial time for (1,1)-polar, also known as split graphs. In this paper, we address the question of distinguishing new classes of graphs admitting polynomial-time solutions for the two problems in question. To this end, we extend the hierarchy of (α,β)-polar graphs and study the computational complexity of the problems on polar graphs of special types.     

Enumerating maximal independent sets with applications to graph colouring

We give tight upper bounds on the number of maximal independent sets of size k (and at least k and at most k) in graphs with n vertices. As an application of the proof, we construct improved algorithms for graph colouring and computing the chromatic number of a graph.     

The number of maximal independent sets in connected graphs

Generalizing a theorem of Moon and Moser, we determine the maximum number of maximal independent sets in a connected graph on n vertices for n sufficiently large, e.g., n > 50.     

An improvement on the maximum number of -dominating independent sets

Erdős and Moser raised the question of determining the maximum number of maximal cliques or, equivalently, the maximum number of maximal independent sets in a graph on vertices. Since then there has been a lot of research along these lines. A -dominating independent set is an independent set such that every vertex not contained in has at least neighbors in . Let denote the maximum number of -dominating independent sets in a graph on vertices, and let . Nagy initiated the study of . In this study, we disprove a conjecture of Nagy using a graph product construction and prove that for any even we have We also prove that for any we have improving the upper bound of Nagy.     

Projectivity and independent sets in powers of graphs

We investigate the relationship between projectivity and the structure of maximal independent sets in powers of circular graphs, Kneser graphs and truncated simplices. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 162–171, 2002     

On independent sets in random graphs

The independence number of a sparse random graph G(n,m) of average degree d = 2m/n is well‐known to be with high probability, with in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size , i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size for any fixed (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point . Roughly speaking, we prove that independent sets of size form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 436–486, 2015     

Reciprocal complementary Wiener numbers of trees, unicyclic graphs and bicyclic graphs

The reciprocal complementary Wiener number of a connected graph G is defined as

where V(G) is the vertex set, d(u,v|G) is the distance between vertices u and v, d is the diameter of G. We determine the trees with the smallest, the second smallest and the third smallest reciprocal complementary Wiener numbers, and the unicyclic and bicyclic graphs with the smallest and the second smallest reciprocal complementary Wiener numbers.     

Constraints on the number of maximal independent sets in graphs

A maximal independent set of a graph G is an independent set that is not contained properly in any other independent set of G. Let i(G) denote the number of maximal independent sets of G. Here, we prove two conjectures, suggested by P. Erdös, that the maximum number of maximal independent sets among all graphs of order n in a family Φ is o(3^n/3) if Φ is either a family of connected graphs such that the largest value of maximum degrees among all graphs of order n in Φ is o(n) or a family of graphs such that the approaches infinity as n → ∞.     

Minimizing the number of independent sets in triangle-free regular graphs

Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin–Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give a new bound on the number of independent sets in triangle-free cubic graphs.     

Maximal independent sets in caterpillar graphs

A caterpillar graph is a tree in which the removal of all pendant vertices results in a chordless path. In this work, we determine the number of maximal independent sets (mis) in caterpillar graphs. For a general graph, this problem is #P—complete. We provide a polynomial time algorithm to generate the whole family of mis in a caterpillar graph. We also characterize the independent graph (intersection graph of mis) and the clique graph (intersection graph of cliques) of complete caterpillar graphs.     

Probabilistic analysis of a parallel algorithm for finding maximal independent sets

We consider a natural parallel version of the classical greedy algorithm for finding a maximal independent set in a graph. This version was studied in Coppersmith, Raghavan, and Tompa and they conjecture there that its expected running time on random graphs of arbitrary edge density of O (log n). We prove that conjecture.     

Estimates of the number of independent sets in graphs with a fixed independence number

Attainable estimates of the number of independent sets in graphs with a given size of the maximal independent set are obtained. Three graph classes—trees, forests, and the class of all graphs—are considered. Extremal graphs are described.     

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

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let G(n,p) denote the set of unicyclic graphs with n vertices and p pendent vertices. In [H. Hua, M. Wang, Unicyclic graphs with given number of pendent vertices and minimal energy, Linear Algebra Appl. 426 (2007) 478-489], Hua and Wang discussed the graphs that have minimal energy in G(n,p), and determined the minimal-energy graphs among almost all different cases of n and p. In their work, certain case of the values was left as an open problem in which the minimal-energy species have to be chosen in two candidate graphs, but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials. This paper aims at solving the problem completely, by using the well-known Coulson integral formula.     

On the spectral radii of unicyclic graphs with fixed matching number

In this paper, the first five sharp upper bounds on the spectral radii of unicyclic graphs with fixed matching number are presented. The first ten spectral radii over the class of unicyclic graphs on a given number of vertices and the first four spectral radii of unicyclic graphs with perfect matchings are also given, respectively.