Independence Polynomial

We consider a relaxation of the concept of well-covered graphs, which are graphs with all maximal independent sets of the same size. The extent to which a graph fails to be well-covered can be measured by its independence gap, defined as the difference between the maximum and minimum sizes of a maximal independent set in G. While the wellcovered graphs are exactly the graphs of independence gap zero, we investigate in this paper graphs of independence gap one, which we also call almost well-covered graphs. Previous works due to Finbow et al. and Barbosa et al. have implications for the structure of almost well-covered graphs of girth at least k for k ∈ {7, 8}. We focus on almost well-covered graphs of girth at least 6. We show that every graph in this class has at most two vertices each of which is adjacent to exactly 2 leaves. We give efficiently testable characterizations of almost well-covered graphs of girth at least 6 having exactly one or exactly two such vertices. Building on these results, we develop a polynomial-time recognition algorithm of almost well-covered {C3, C4, C5, C7}-free graphs.

We study graphs in which the maximum and the minimum sizes of a maximal independent set differ by exactly one. We call these graphs almost well-covered, in analogy with the class of well-covered graphs, in which all maximal independent sets have the same size. A characterization of graphs of girth at least $8$ having exactly two different sizes of maximal independent sets due to Finbow, Hartnell, and Whitehead implies a polynomial-time recognition algorithm for the class of almost well-covered graphs of girth at least $8$. We focus on the structure of almost well-covered graphs of girth at least $6$. We give a complete structural characterization of a subclass of the class of almost well-covered graphs of girth at least $6$, a polynomially testable characterization of another one, and a polynomial-time recognition algorithm of almost well-covered $\{C_3,C_4,C_5,C_7\}$-free graphs.

Given a hypergraph H and a function f : V (H) -→ N, we say that H is f -choosable if there exists a proper vertex colouring φ of H such that φ The class of sc-greedy hypergraphs is closed under the union of hypergraphs having at most one vertex in common. In this paper we consider sc-greediness of the union of hypergraphs having two vertices in common. We investigate this operation when one of the arguments is an arbitrary sc-greedy hypergraph while the second one is a hyperpath. Our research is motivated by the possibility of obtaining improved bounds on the sumchoice-number of graphs and new applications to the resource allocation problems in computer systems.

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while µ(G) is the cardinality of a maximum matching. If s k is the number of independent sets of size k in G, then and Harary, 1986). If s j = s α-j for all 0 ≤ j ≤ α/2 , then I(G; x) is called symmetric (or palindromic). It is known that the graph G • 2K 1 , obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial . In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer k ≤ µ (G), one can build a graph H, such that: G is a subgraph of H, I (H; x) is symmetric, and I (G • 2K 1 ; x) = (1 + x) k • I (H; x).

If s k denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then . A graph G is very well-covered if it has no isolated vertices, its order equals 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M. D. . 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 conditions, any well-covered graph equals G * for some G (Finbow, . The root of the smallest modulus of the independence polynomial of any graph is real . The location of the roots of the independence polynomial in the complex plane, and the multiplicity of the root of the smallest modulus are investigated in a number of articles In this paper we establish formulae connecting the coefficients of I(G; x) and I(G * ; x), which allow us to show that the number of roots of I(G; x) is equal to the number of roots of I(G * ; x) different from -1, which appears as a root of multiplicity α(G * ) -α(G) for I(G * ; x). We also prove that the real roots of I(G * ; x) are in [-1, -1/2α(G * )), while for a general graph of order n we show that its roots lie in |z| > 1/(2n -1). Hoede and Li (1994) posed the problem of finding graphs that can be uniquely defined by their clique polynomials (clique-unique graphs). Stevanovic (1997) proved that threshold graphs are clique-unique. Here, we demonstrate that the independence polynomial distinguishes well-covered spiders (K * 1,n , n ≥ 1) among well-covered trees.

Abstract. The stability number of the graph G, denoted by α (G), is the cardinality of a maximum stable set of G. In this paper we characterize the square-stable graphs, ie, the graphs enjoying the property α (G)= α (G 2), where G 2 is the graph with the same vertex set as in G, and an edge of G 2 is joining two distinct vertices, whenever the distance between them in G is at most 2. We show that every square-stable graph is well-covered, and wellcovered trees are exactly the square-stable trees. Keywords: stable set, square-stable ...

Throughout this paper G = (V,E) is a simple (ie, a finite, undirected, loopless and without multiple edges) graph with vertex set V = V (G) and edge set E = E(G). If X ⊂ V , then G[X] is the subgraph of G spanned by X. By G − W we mean the subgraph G[V − W], if W ⊂ V (G). We also denote by G − F the partial subgraph of G obtained by deleting the edges of F, for F ⊂ E(G), and we write shortly G − e, whenever F = {e}. The neighborhood of a vertex v ∈ V is the set NG(v) = {w : w ∈ V and vw ∈ E}, and NG[v] = NG(v) ∪ {v}; if there is ambiguity on G, we use N(v) ...

ABSTRACT. A stable (or independent) set in a graph is a set of pairwise non-adjacent vertices. The stability number α(G) is the size of a maximum stable set in the graph G. The independence polynomial of G is defined by I(G; x) = s0 + s1x + s2x2 + ... + sαxα, α = α(G), where sk equals the number of stable sets of cardinality k in G (I. Gutman and F. Harary, 1983). In this paper, we build a family of graphs whose independence polynomials are palindromic and unimodal. We conjecture that all these polynomials are also log-concave. ... Received: 21.11.2006; In ...

Abstract: 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). JI Brown, K. Dilcher and RJ Nowakowski (2000) conjectured that the independence polynomial of a well-covered graph $ G $(ie, 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 $- ...

A graph with at most two vertices of the same degree is known as antiregular [ Merris, R., Antiregular graphs are universal for trees, Publ. Electrotehn. Fak. Univ. Beograd, Ser. Mat. 14 (2003) 1-3], maximally nonregular [Zykov, A. A., Fundamentals of graph theory, BCS Associates, Moscow, 1990] or quasiperfect [ Behzad, M. and Chartrand, D. M., No graph is perfect, Amer. Math. Monthly 74 (1967), 962-963]. If sk is the number of independent sets of cardinality k in a graph G, then I(G; x) = s0 +s1x+...+sαx α is the independence polynomial of G [ Gutman, I. and Harary, F., Generalizations of the matching polynomial, Utilitas Mathematica 24 (1983), 97-106] , where α = α(G) is the size of a maximum independent set. In this paper we derive closed formulas for the independence polynomials of antiregular graphs. It results in proving that every antiregular graph is uniquely defined by its independence polynomial within the family of threshold graphs. Moreover, the independence polynomial o...

In [18], Farrell and Whitehead investigate circulant graphs that are uniquely characterized by their matching and chromatic polynomials (i.e., graphs that are "matching unique" and "chromatic unique"). They develop a partial classification theorem, by finding all matching unique and chromatic unique circulants on n vertices, for each n ≤ 8. In this paper, we explore circulant graphs that are uniquely characterized by their independence polynomials. We obtain a full classification theorem by proving that a circulant is independence unique iff it is the disjoint union of isomorphic complete graphs.

A stable set in a graph G is a set of mutually non-adjacent vertices, α(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T , of order n ≥ 2, any stable set of size ≥ n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. We give a new proof for a theorem of Hopkins and Staton [5] characterizing strong unique trees. Using this result we show that if {A, B} is the bipartition of a tree T and S is a stable set with |S| > min{|A| , |B|}, then S contains at least a pendant vertex. Our main finding is the theorem claiming that if T is a tree of order n ≥ 2 that does not own a perfect matching (i.e., 2α(T) > n), then at least two pendant vertices an even distance apart belong to core(T). While it is known that if G is a connected bipartite graph of order n ≥ 2, then |core(G)| = 1 (see Levit, Mandrescu [7]), our new statement reveals an additional structure of the intersection of all maximum stable sets of a tree. The above assertions give refining of one assertion of Hammer, Hansen and Simeone [4] stating that if a graph G is of order less than 2α(G), then core(G) is non-empty, and also of a result of Jamison [6], Gunter, Hartnel and Rall [3], and Zito [10], saying that for a tree T of order at least two, |core(T)| = 1.

If s k denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then I(G; x) = α(G) k=0 s k x k is 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 2α(G) and it is well-covered (i.e., all its maximal independent sets are of the same size, M. D. 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 conditions, any well-covered graph equals G * for some G (Finbow, Hartnell and Nowakowski, 1993). The root of the smallest modulus of the independence polynomial of any graph is real (Brown, Dilcher, and Nowakowski, 2000). The location of the roots of the independence polynomial in the complex plane, and the multiplicity of the root of the smallest modulus are investigated in a number of articles In this paper we establish formulae connecting the coefficients of I(G; x) and I(G * ; x), which allow us to show that the number of roots of I(G; x) is equal to the number of roots of I(G * ; x) different from −1, which appears as a root of multiplicity α(G *) − α(G) for I(G * ; x). We also prove that the real roots of I(G * ; x) are in [−1, −1/2α(G *)), while for a general graph of order n we show that its roots lie in |z| > 1/(2n − 1). Hoede and Li (1994) posed the problem of finding graphs that can be uniquely defined by their clique polynomials (clique-unique graphs). Stevanovic (1997) proved that threshold graphs are clique-unique. Here, we demonstrate that the independence polynomial distinguishes well-covered spiders (K * 1,n , n ≥ 1) among well-covered trees.

Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of the computing, social, and natural sciences. The fastest growing area within graph theory is the study of domination and Independence numbers. Domination number is the cardinality of a minimum dominating set of a graph. Independence number is the maximal cardinality of an independent set of vertices of a graph. The concept of Fibonacci numbers of graphs was first introduced by Prodinger and Tichy in 1982. The Fibonacci numbers of a graph is the number of independent vertex subsets. In this paper, introduce the identities of domination, independence and Fibonacci numbers of graphs containing vertex-disjoint cycles and edge-disjoint cycles.

It is shown that the Shields-Harary index of vulnerability of the complete bipartite graph K m,n , with respect to the cost function f (x) = 1 − x, 0 x 1, is m, if n m + 2 √ m, and 1 n+1 (n+m) 2 4 , if m n < m + 2 √ m. It follows that the Shields-Harary number of K m,n with respect to any concave continuous cost function f on [0, 1] satisfying f (1) = 0 is mf (0), if n m + 2 √ m, and between 1 n+1 (n+m) 2 4 f (0) and mf (0), if m n < m + 2 √ m.

We consider the synthesis of a fixed order or fixed structure multivariable feedback controller C, parametrized by a design vector x, for a plant P, containing a vector p of uncertain parameters. The characteristic polynomials of such systems contain coefficients which depend polynomially on x and p. Using results on sign definite decomposition we develop a 4-polynomial stability test that gives a sufficient condition for stability of the family of closed loop systems that result when x and p vary over prescribed boxes. This test is reminiscent of Kharitonov¿s theorem, even though the family of polynomials considered here is certainly not restricted to be interval or even convex. Moreover this result is tight in the sense that the test does reduce to Kharitonov¿s theorem for the special case of interval polynomials. Using this criterion recursively and modularly we design an algorithm to determine sets of controllers that stabilize the family of uncertain plants. Examples and futu...

An independent set of a graph is a set of pairwise non-adjacent vertices while the independence number is the maximum cardinality of an independent set in the graph. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called n-th central if the vertices are elements of G and two distinct vertices are adjacent if they are elements in the n-th term of the upper central series of G. In this research, the independence polynomial of the n-th central graph is found for some dihedral groups. The results are computed by using the definition of independence polynomial and also by using the independence polynomial of the union of complete graphs.

The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\cH$-combinations of impartial games for any hypergraph $\cH$. In particular, we introduce the game $NIM_\cH$ which is the $\cH$-combination of single pile $NIM$ games. An impartial game is called SG decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce the $\cH$-combination of SG decreasing games to $NIM_\cH$. We call $\cH$ a Tetris hypergraph if $NIM_\cH$ is SG decreasing. We provide some necessary and some sufficient conditions for a hypergraph to be Tetris.

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while µ(G) is the cardinality of a maximum matching. If s k is the number of independent sets of size k in G, then I(G; x) = s 0 + s 1 x + s 2 x 2 + ... + s α x α , α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If s j = s α−j for all 0 ≤ j ≤ α/2 , then I(G; x) is called symmetric (or palindromic). It is known that the graph G • 2K 1 , obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer k ≤ µ (G), one can build a graph H, such that: G is a subgraph of H, I (H; x) is symmetric, and I (G • 2K 1 ; x) = (1 + x) k • I (H; x).

The Sprague-Grundy (SG) theory reduces the sum of impartial games to the classical game of $NIM$. We generalize the concept of sum and introduce $\cH$-combinations of impartial games for any hypergraph $\cH$. In particular, we introduce the game $NIM_\cH$ which is the $\cH$-combination of single pile $NIM$ games. An impartial game is called SG decreasing if its SG value is decreased by every move. Extending the SG theory, we reduce the $\cH$-combination of SG decreasing games to $NIM_\cH$. We call $\cH$ a Tetris hypergraph if $NIM_\cH$ is SG decreasing. We provide some necessary and some sufficient conditions for a hypergraph to be Tetris.

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G)+mu(G) equals its order, then G is a Konig-Egervary graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality alpha(G)=alpha(G^2), where G^2 denotes the second power of G. In particular, we show that a Konig-Egervary graph is square-stable if and only if it has a perfect matching consisting of pendant edges, and in consequence, we deduce that well-covered trees are exactly the square-stable trees.

A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_k is the number of independent sets of cardinality k in a graph G, then I(G;x) = s_0 + s_1x + ... + s_alphax^alpha is the independence polynomial of G (Gutman, Harary 1983), where alpha = alpha(G) is the size of a maximum independent set. In this paper we derive closed formulae for the independence polynomials of antiregular graphs. In particular, we deduce that every antiregular graph A is uniquely defined by its independence polynomial I(A;x), within the family of threshold graphs. Moreover, I(A;x) is logconcave with at most two real roots, and I(A;-1) belongs to -1,0.

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_1x+...+s_alphax^alpha is the independence polynomial of G. In this paper we provide an elementary proof of the inequality claiming that the absolute value of I(G;-1) is not greater than 2^phi(G), for every graph G, where phi(G) is its decycling number.

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_1x+...+s_alphax^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 &gt; 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.

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(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) = s0 + s1x + s2x 2 + ... + sαx α , α = α (G) , is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If sα−i = f (i) • si holds for every i ∈ {0, 1, ..., ⌊α/2⌋}, then I(G; x) is called fsymmetric (f-palindromic). If f (i) = 1, i ∈ {0, 1, ..., ⌊α/2⌋}, then I(G; x) is called 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 this paper we show that if H is a graph with p vertices, q edges, and α (H) = 2, then I (G • H; x) is f-symmetric, where f (i) = p (p − 1) 2 − q α 2 −i , 0 ≤ i ≤ α = α (G • H). In particular, if H = Kr − e, r ≥ 2, we show that I (G • H; x) is symmetric and unimodal, with a unique mode. This finding generalizes results due to Stevanović [22] and Mandrescu [21] claiming that I (G • (K2 − e) ; x) = I (G • 2K1; x) is symmetric and unimodal for every graph G.

An independent set of a graph is a set of pairwise non-adjacent vertices while the independence number is the maximum cardinality of an independent set in the graph. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called n-th central if the vertices are elements of G and two distinct vertices are adjacent if they are elements in the n-th term of the upper central series of G. In this research, the independence polynomial of the n-th central graph is found for some dihedral groups. The results are computed by using the definition of independence polynomial and also by using the independence polynomial of the union of complete graphs.

An independent set of a graph is a set of pairwise non-adjacent vertices while the independence number is the maximum cardinality of an independent set in the graph. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called n-th central if the vertices are elements of G and two distinct vertices are adjacent if they are elements in the n-th term of the upper central series of G. In this research, the independence polynomial of the n-th central graph is found for some dihedral groups. The results are computed by using the definition of independence polynomial and also by using the independence polynomial of the union of complete graphs.

A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_{k} is the number of independent sets of cardinality k in a graph G, then I(G;x) = s_{0} + s_{1}x + ... + s_{alpha}x^{alpha} is the independence polynomial of G (Gutman, Harary 1983), where alpha = alpha(G) is the size of a maximum independent set. In this paper we derive closed formulae for the independence polynomials of antiregular graphs. In particular, we deduce that every antiregular graph A is uniquely defined by its independence polynomial I(A;x), within the family of threshold graphs. Moreover, I(A;x) is logconcave with at most two real roots, and I(A;-1) belongs to {-1,0}.

Throughout this paper G = (V,E) is a simple (ie, a finite, undirected, loopless and without multiple edges) graph with vertex set V = V (G) and edge set E = E(G). If X ⊂ V , then G[X] is the subgraph of G spanned by X. By G − W we mean the subgraph G[V − W], if W ⊂ V (G). We also denote by G − F the partial subgraph of G obtained by deleting the edges of F, for F ⊂ E(G), and we write shortly G−e, whenever F = {e}. The neighborhood of a vertex v ∈ V is the set NG(v) = {w : w ∈ V and vw ∈ E}, and NG[v] = NG(v) ∪ {v}; if there is no ambiguity on G, we use N(v) ...

The stability number of a graph G, denoted by α(G), is the cardinality of a maximum stable set, and µ(G) is the cardinality of a maximum matching in G. If α(G) + µ(G) equals its order, then G is a König-Egerváry graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality α(G) = α(G 2), where G 2 denotes the second power of G. In particular, we show that a König-Egerváry graph is square-stable if and only if it has a perfect matching consisting of pendant edges, and in consequence, we deduce that well-covered trees are exactly the square-stable trees.

The square of a graph G is the graph G 2 with the same vertex set as in G, and an edge of G 2 is joining two distinct vertices, whenever the distance between them in G is at most 2. G is a square-stable graph if it enjoys the property α(G) = α(G 2), where α(G) is the size of a maximum stable set in G. In this paper we show that G 2 is a König-Egerváry graph if and only if G is a square-stable König-Egerváry graph.

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G an $\alpha $-square-stable graph, shortly square-stable, if alpha(G) = alpha(G*G), where G*G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann. In this paper we obtain several new characterizations of square-stable graphs. We also show that G is an square-stable Koenig-Egervary graph if and only if it has a perfect matching consisting of pendant edges. Moreover, we find that well-covered trees are exactly square-stable trees. To verify this result we give a new proof of one Ravindra&#x27;s theorem describing well-covered trees.

The concept of the matching polynomial of a graph, introduced by Farrell in 1979, has received considerable attention and research. In this paper, we generalize this concept and introduce the matching polynomial of hypergraphs. A recurrence relation of the matching polynomial of a hypergraph is obtained. The exact matching polynomials of some special hypergraphs are given. Further, we discuss the zeros of matching polynomials of hypergraphs.