A Family of Well-Covered Graphs with Unimodal Independence Polynomials
Vadim E. Levit, Eugen Mandrescu
备注：
This paper has been presented at Thirty-fourth Southeastern International Conference on Combinatorics, Graph Theory, and Computing, March 2003, Florida Atlantic University, USA; 13 pages; 2 figures
If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that the independence polynomial of a well-covered graph $G$ (i.e., a graph whose all maximal independent sets are of the same size) is unimodal, that is, there exists an index $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the other part of coefficients is non-increasing.
T. S. Michael and N. Traves (2002) provided examples of well-covered graphs whose independence polynomials are not unimodal.
A. Finbow, B. Hartnell and R. J. Nowakowski (1993) proved that under certain conditions, any well-covered graph equals G* for some $G$, where G* is the graph obtained from $G$ by appending a single pendant edge to each vertex of $G$. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös (1987) asked whether for trees the independence polynomial is unimodal. V. E. Levit and E. Mandrescu (2002) validated the unimodality of the independence polynomials of some well-covered trees (e.g., $P_{n}^{*},K_{1,n}^{*}$, where $P_{n}$ is the path on $n$ vertices and $K_{1,n}$ is the $n$-star graph).
In this paper we show that for any graph $G$ with the stability number alpha(G) < 5, the independence polynomial of G* is unimodal.
In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems.
The independence polynomial $i(G,x)$ of a graph $G$ is the generating function of the numbers of independent sets of each size. A graph of order $n$ is very well-covered if every maximal independent set has size $n/2$. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of…
The independence polynomial $I(G, x)$ of a graph $G$ is the polynomial in variable $x$ in which the coefficient $a_n$ on $x^n$ gives the number of independent subsets $S \subseteq V(G)$ of vertices of $G$ such that $|S| = n$. $I(G, x)$ is unimodal if there is an index $μ$ such that that $a_0 \leq a_1 \leq$...$\leq a_{μ-1} \leq a_μ \geq a_{μ+1} \geq$...$\geq a_{d-1} \geq a_d$ While the independence polynomials of many families of graphs with highly regular structure are known to be unimodal, little is known about less regularly structured graphs. We analyze the independence polynomials of a lar…
The number of stable sets of cardinality $k$ in graph $G$ is the $k$-th coefficient of the independence polynomial of $G$ (I. Gutman and F. Harary, 1983). In 1990, Y. O. Hamidoune proved that for any claw-free graph, its independence polynomial is unimodal, i.e., there exists a coefficient $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the second part of coefficients is non-increasing. Y. Alavi, P. J. Malde, A. J. Schwenk and P. Erdös (1987) asked whether for trees (or perhaps forests) the independence polynomial is unimodal. J. I. Brown…
A graph $G$ is well-covered if all its maximal stable sets have the same size, denoted by alpha(G) (M. D. Plummer, 1970). If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). J. I. Brown, K. Dilcher and R. J. Nowakowski (2000) conjectured that I(G;x) is unimodal (that is, there exists an index $k$ such that the part of the sequence of coefficients from the first to $k$-th is non-decreasing while the other part of coefficients is non-incr…
If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Let $a$ be the size of a maximum stable set. Alavi, Malde, Schwenk and Erdos (1987)conjectured that I(T,x) is unimodal for any tree T, while, in general, they proved that for any permutation $p$ of {1,2,...,a} there is a graph such that s_{p(1)}<s_{p(2)}<...<s_{p(a)}. Brown, Dilcher and Nowakowski (2000) conjectured that I(G;x) is unimodal for any well-covered graph. Michael and Tr…
Given two graphs $G$ and $H$, assume that $\mathscr{C}=\{C_1,C_2,\ldots, C_q\}$ is a clique cover of $G$ and $U$ is a subset of $V(H)$. We introduce a new graph operation called the clique cover product, denoted by $G^{\mathscr{C}}\star H^U$, as follows: for each clique $C_i\in \mathscr{C}$, add a copy of the graph $H$ and join every vertex of $C_i$ to every vertex of $U$. We prove that the independence polynomial of $G^{\mathscr{C}}\star H^U$ $$I(G^{\mathscr{C}}\star H^U;x)=I^q(H;x)I(G;\frac{xI(H-U;x)}{I(H;x)}),$$ which generalizes some known results on independence polynomials of corona and …
Let $I(G;x)$ denote the independence polynomial of a graph $G$. In this paper we study the unimodality properties of $I(G;x)$ for some composite graphs $G$. Given two graphs $G_1$ and $G_2$, let $G_1[G_2]$ denote the lexicographic product of $G_1$ and $G_2$. Assume $I(G_1;x)=\sum_{i\geq0}a_ix^i$ and $I(G_2;x)=\sum_{i\geq0}b_ix^i$, where $I(G_2;x)$ is log-concave. Then we prove (i) if $I(G_1;x)$ is log-concave and $(a^2_i-a_{i-1}a_{i+1})b^2_1\geq a_ia_{i-1}b_2$ for all $1\leq i \leq α(G_1)$, then $I(G_1[G_2];x)$ is log-concave; (ii) if $a_{i-1}\leq b_1a_i$ for $1\leq i\leq α(G_1)$, then $I(G_1[…
An independent set in a graph is a set of pairwise non-adjacent vertices. The independence number $α{(G)}$ is the size of a maximum independent set in the graph $G$. The independence polynomial of a graph is the generating function for the sequence of numbers of independent sets of each size. In other words, the $k$-th coefficient of the independence polynomial equals the number of independent sets comprised of $k$ vertices. For instance, the degree of the independence polynomial of the graph $G$ is equal to $α{(G)}$. In 1987, Alavi, Malde, Schwenk, and Erd{ö}s conjectured that the independenc…
A graph $G$ is a fractional $(a,b,k)$-critical covered graph if $G-U$ is a fractional $[a,b]$-covered graph for every $U\subseteq V(G)$ with $|U|=k$, which is first defined by Zhou, Xu and Sun (S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional $(a,b,k)$-critical covered graphs, Information Processing Letters, DOI: 10.1016/j.ipl.2019.105838). Furthermore, they derived a degree condition for a graph to be a fractional $(a,b,k)$-critical covered graph. In this paper, we gain an independence number and connectivity condition for a graph to be a fractional $(a,b,k)$-critical covered graph an…
Unimodality of Independence Polynomials for Sufficiently Large Forests
Duke University（杜克大学）；Harvard T.H. Chan School of Public Health（哈佛陈曾熙公共卫生学院）；Hebrew University（希伯来大学）；Universidad de Valladolid（巴利亚多利德大学）；University of Hawai‘i at Mānoa（夏威夷大学马诺阿分校）
If for any k the k-th coefficient of a polynomial I(G;x)is equal to the number of stable sets of cardinality k in graph G, then it is called the independence polynomial of G (Gutman and Harary, 1983). A graph G is very well-covered (Favaron, 1982) if it has no isolated vertices, its order equals 2*alpha(G), where alpha(G) is the size of a maximum stable set, and it is well-covered (i.e., all its maximal independent sets are of the same size, Plummer, 1970). For instance, appending a single pendant edge to each vertex of G yields a very well-covered graph, which we denote by G*. Under certain c…
We study the independence complexes of families of well-covered circulant graphs discovered by Boros-Gurvich-Milanič, Brown-Hoshino, and Moussi. Because these graphs are well-covered, their independence complexes are pure simplicial complexes. We determine when these pure complexes have extra combinatorial (e.g. vertex decomposable, shellable) or topological (e.g. Cohen-Macaulay, Buchsbaum) structure. We also provide a table of all well-covered circulant graphs on 16 or less vertices, and for each such graph, determine if it is vertex decomposable, shellable, Cohen-Macaulay, and/or Buchsbaum. …
If for any $k$ the $k$-th coefficient of a polynomial $I(G;x)$ is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdos (1987) conjectured that $I(G;x)$ is unimodal, whenever $G$ is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that $I(G;x)$ is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with $a(G)$ > 3 ($a(G)$ is the size of a maximum stable set of the graph $G$), whi…
A graph $G$ is well-covered if all maximal independent sets of $G$ have the same cardinality. In 1992 Topp and Volkmann investigated the structure of well-covered graphs that have nontrivial factorizations with respect to some of the standard graph products. In particular, they showed that both factors of a well-covered direct product are also well-covered and proved that the direct product of two complete graphs (respectively, two cycles) is well-covered precisely when they have the same order (respectively, both have order 3 or 4). Furthermore, they proved that the direct product of two well…
The independence polynomial of a graph $G$, denoted $I(G,x)$, is the generating polynomial for the number of independent sets of each size. The roots of $I(G,x)$ are called the \textit{independence roots} of $G$. It is known that for every graph $G$, the independence root of smallest modulus, denoted $ξ(G)$, is real. The relation $\preceq$ on the set of all graphs is defined as follows, $H\preceq G$ if and only if $I(H,x)\ge I(G,x)\text{ for all }x\in [ξ(G),0].$ We find the maximum and minimum connected unicyclic and connected well-covered unicyclic graphs of a given order with respect to $\pr…
A graph $G$ is {\em well-covered} if every maximal independent set has the same cardinality $q$. Let $i_k(G)$ denote the number of independent sets of cardinality $k$ in $G$. Brown, Dilcher, and Nowakowski conjectured that the independence sequence $(i_0(G), i_1(G), \ldots, i_q(G))$ was unimodal for any well-ordered graph $G$ with independence number $q$. Michael and Traves disproved this conjecture. Instead they posited the so-called ``Roller Coaster" Conjecture: that the terms \[ i_{\left\lceil\frac{q}2\right\rceil}(G), i_{\left\lceil\frac{q}2\right\rceil+1}(G), \ldots, i_q(G) \] could be in…
The $k$-token graph $T_k(G)$ is the graph whose vertices are the $k$-subsets of vertices of a graph $G$, with two vertices of $T_k(G)$ adjacent if their symmetric difference is an edge of $G$. We explore when $T_k(G)$ is a well-covered graph, that is, when all of its maximal independent sets have the same cardinality. For bipartite graphs $G$, we classify when $T_k(G)$ is well-covered. For an arbitrary graph $G$, we show that if $T_2(G)$ is well-covered, then the girth of $G$ is at most four. We include upper and lower bounds on the independence number of $T_k(G)$, and provide some families of…
A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer, 1970). If G is a well-covered graph, has at least two vertices, and G-v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples, 1975). We call G a λ-quasi-regularizable graph if λ |S| =< |N(S)| for every independent set S of G. The independence polynomial I(G;x) is the generating function of independent sets in a graph G (Gutman & Harary, 1983). The Roller-Coaster Conjecture (Michael & Travis, 2003), saying that for every permutation σ of the set {(α/2),...,α} there exists a…
A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number. We answer a question of Topp and Volkmann and prove that if the Cartesian product of two graphs is well-covered, then at least one of them must be well-covered.
An independent set in a graph is a set of pairwise non-adjacent vertices, and a(G) is the size of a maximum independent set in the graph G. If s_{k} is the number of independent sets of cardinality k in G, then I(G;x)=s_0+s_1*x+s_2*x^2+...+s_a*x^a,a=a(G), is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If s_{a-i}=f(i)*s_{i} holds for every i, then I(G;x) is called f-symmetric (f-palindromic). If f(i)=1, then I(G;x) is symmetric (palindromic). The corona of the graphs G and H is the graph G*H obtained by joining each vertex of G to all the vertices of a copy of H. In…
The independence gap of a graph was introduced by Ekim et al. (2018) as a measure of how far a graph is from being well-covered. It is defined as the difference between the maximum and minimum size of a maximal independent set. We investigate the independence gap of a graph from structural and algorithmic points of view, with a focus on classes of perfect graphs. Generalizing results on well-covered graphs due to Dean and Zito (1994) and Hujdurović et al. (2018), we express the independence gap of a perfect graph in terms of clique partitions and use this characterization to develop a polynomi…
If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we prove that: 1. I(T;-1) equels either -1 or 0 or 1 for every tree T; 2. I(G;-1)=0 for every connected well-covered graph G of girth > 5, non-isomorphic to C_{7} or K_{2}; 3. the absolute value of I(G;-1) is not greater than 2^nu(G), for every graph G, where nu(G) is its cyclomatic number.
A broadcast on a nontrivial connected graph G is a function f from the vertices of G to the non-negative integers such that f(v) does not exceed e(v) (the eccentricity of v) for each vertex v. If G is disconnected, we define a broadcast on G as the union of broadcasts on its components. In a search for the best way to generalise the concept of independent sets in graphs to independent broadcasts, there are several ways to look at an independent set X of a graph G. One way is from the point of view of the vertices in X: no two vertices are adjacent -- the usual definition. Another way is from t…
The linear complementarity problem is a continuous optimization problem that generalizes convex quadratic programming, Nash equilibria of bimatrix games and several such problems. This paper presents a continuous optimization formulation for the weighted independence number of a graph by characterizing it as the maximum weighted $\ell_1$ norm over the solution set of a linear complementarity problem (LCP). The minimum $\ell_1$ norm of solutions of this LCP is a lower bound on the independent domination number of the graph. Unlike the case of the maximum $\ell_1$ norm, this lower bound is in ge…
Planar ternary graphs, flag spheres, and Delannoy polynomials
University of Kansas（堪萨斯大学）；University of Vermont（佛蒙特大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进研究所数学系）；Youngstown State University（扬斯敦州立大学）；Georgia Institute of Technology（佐治亚理工学院）
Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $Δ$ be its independence complex. We describe the well-covered circulant graphs with 2-dimensional $Δ$ and construct an infinite family of vertex-decomposable circulant graphs within this family.
A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs. In this paper we obtain several structural and algori…
A Ridge-Saturation Characterization of $α$-Critical $\mathbf {W}_p$ Graphs
Hanoi University of Science and Technology（河内科技大学）；Ariel University（阿里埃勒大学）；Holon Institute of Technology（霍隆理工学院）；Universidad Nacional de San Luis（圣路易斯国立大学）
A circulant is a Cayley graph over a cyclic group. A well-covered graph is a graph in which all maximal stable sets are of the same size, or in other words, they are all maximum. A CIS graph is a graph in which every maximal stable set and every maximal clique intersect. It is not difficult to show that a circulant G is a CIS graph if and only if G and its complement are both well-covered and the product of the independence and the clique numbers of G is equal to the number of vertices. It is also easy to demonstrate that both families, the circulants and the CIS graphs, are closed with respec…
