Optimal Graphs for Independence and k -Independence Polynomials
OOptimal Graphs for Independence and k -Independence Polynomials J.I. Brown , D. Cox September 19, 2018 Department of Mathematics and Statistics, Dalhousie University,Halifax, Canada B3H 4R2, [email protected] Department of Mathematics, Mount Saint Vincent University,Halifax, Canada B3M 2J6, [email protected] 4, 2017
Abstract
The independence polynomial I ( G, x ) of a finite graph G is the gen-erating function for the sequence of the number of independent sets ofeach cardinality. We investigate whether, given a fixed number of ver-tices and edges, there exists optimally-least ( optimally-greatest ) graphs,that are least (respectively, greatest) for all non-negative x . Moreover, webroaden our scope to k -independence polynomials, which are generatingfunctions for the k -clique-free subsets of vertices. For k ≥
3, the resultscan be quite different from the k = 2 (i.e. independence) case. Given a property of subsets of the vertex or edge set – such as independent, com-plete, dominating for vertices and matching for edges – one is often interested inmaximizing or minimizing the size of the set in a given graph G . However, onecan get a much more nuanced study of the subsets by studying the number ofsuch sets of each cardinality in G , and in this guise, one often encapsulates thesequence by forming a generating function. Independence, clique, dominatingand matching polynomials have all arisen and been studied in this setting.In all cases, the generating polynomial f ( G, x ) is naturally a function onthe domain [0 , ∞ ). If the number of vertices n and edges m are fixed, one canask whether there exists an extremal graph. Let S n,m denote the set of simplegraphs of order n ( n vertices) and size m ( m edges). Let H ∈ S n,m . H is optimally-greatest (or optimally-least ) if f ( H, x ) ≥ f ( G, x ) ( f ( H, x ) ≤ f ( G, x ),respectively) for all graphs G ∈ S n,m and all x ≥ x ≥
0, of course, there is such a graph H , as the number of graphs of order n and size m is finite, but we are interested in uniformly optimal graphs).1 a r X i v : . [ m a t h . C O ] O c t uch questions (related to simple substitutions of generating polynomials)have attracted considerable attention in the areas of network reliability [2–4, 7,10, 14] and chromatic polynomials [15, 16].Here we consider optimality for independence polynomials I ( G, x ) = (cid:88) j i j x j , and a broad generalization, for fixed k ≥
2, to k -independence polynomials I k ( G, x ) = (cid:88) j i k,j x j , where i k,j is the number of subsets of the vertex set of size j that induce a k -clique-free subgraph (that is, the induced subgraph contains no completesubgraph of order k ). Clearly independence polynomials are precisely the 2-independence polynomials, while the 3-independence polynomials are generatingfunctions for the numbers of triangle-free induced subgraphs.In this paper, we look at the optimality of independence polynomials andmore generally k -independence polynomials. In the case of the former, optimally-greatest graphs always exist for independence polynomials, but we do not knowwhether optimally-least graphs necessarily exist for independence polynomialsas well (although we will prove for some n and m they do). In contrast, we shallshow that for k ≥
3, optimality behaves quite differently, in that for some n and m , optimally-least and optimally-greatest graphs for k -independence polynomi-als do not exist. Our first result shows that for independence polynomials, optimally-greatestgraphs indeed do always exist.
Theorem 1
For all n ≥ and all m ∈ { , . . . , (cid:0) n (cid:1) } , an optimally-greatest graphalways exists. Proof : A key observation is that a sufficient condition for H to be optimally-greatest (or optimally-least) for independence polynomials is that i ( H, x ) = (cid:80) i j ( H ) x j is coefficient-wise greatest , that is, for all other graphs G of thesame order and size, the generating polynomials f ( G, x ) = (cid:80) i j ( G ) x j satisfies i j ( H ) ≥ i j ( G ) for all j (respectively, i j ( H ) ≤ i j ( G )). (The coefficient-wisecondition is not, in general, necessary for optimality as, for example, 5 x + x +5 ≥ x + 4 x + 1 for x ≥
0, but clearly 5 x + x + 5 is not coefficient-wise greater thanor equal to x + 4 x + 1.) 2onsider the following graph construction. For a given n and m , take afixed linear order (cid:22) of the vertices, v n (cid:22) v n − (cid:22) · · · (cid:22) v , and select the m largest edges in lexicographic order. We will denote this graph as G n,m, (cid:22) (itis dependent on the linear order, but of course all such graphs are isomorphic).It was shown in [8] that G n,m, (cid:22) is the graph of order n and size m with themost number of independent sets of size j , for all j ≥
0. It follows that theindependence polynomial for G n,m, (cid:22) is coefficient-wise optimally-greatest forindependence polynomials, for all graphs of order n and size m . (cid:3)(cid:3) Moreover, we can compute the independence polynomial of this optimally-greatest graph. To do so, we need the following well-known recursion to computethe independence polynomial of a graph. Let G be a graph with v ∈ V ( G ). Then I ( G, x ) = xI ( G − N [ v ] , x ) + I ( G − v, x ) (1)where N [ v ] is the closed neighbourhood of v and G − v is G with v removed.For graph G n,m, (cid:22) with m < (cid:0) n (cid:1) , write m = ( n − (cid:96) ) + ( n − (cid:96) + 1) + . . . ( n −
1) + j with j ∈ { , . . . , n − (cid:96) − } . Since the edges are added in lexicographic order,there will be k vertices that have degree n − j + (cid:96) .The remaining n − (cid:96) − v be the vertexof degree j + (cid:96) . Using vertex, v and recursion (1), we obtain our result, I ( G n,m, (cid:22) , x ) = I ( G n,m, (cid:22) − v, x ) + xI ( G n,m, (cid:22) − N [ v ] , x )= I ( K (cid:96) ∪ K n − (cid:96) − , x ) + I ( K (cid:96) ∪ K n − (cid:96) − j − , x )= (1 + (cid:96)x )(1 + x ) n − (cid:96) − + x (1 + (cid:96)x )(1 + x ) n − (cid:96) − j − = (1 + (cid:96)x )(1 + x ) n − (cid:96) − j − (cid:0) (1 + x ) j + x (cid:1) . Of course, for m = (cid:0) n (cid:1) , G n,m, (cid:22) = K n and so I ( G n,m, (cid:22) , x ) = 1 + nx .It is instructive that this result can also be proved purely algebraically, andwe devote the remainder of this section to do so. (We refer the reader [6] foran introduction to complexes and their connection to commutative algebra.)The independence complex of graph G of order n and size m , ∆ ( G ), has as itsfaces the independent sets of G (these are obviously closed under containment,and hence a complex). The f -vector of ∆ ( G ) is (1 , n, (cid:0) n (cid:1) − m, f , . . . , f β ),where f i is the number of faces of cardinality i in the complex (and β is theindependence number of G , which is the same as the dimension of the complex).We will show that we can maximize the independence polynomial on [0 , ∞ ) bymaximizing ( simultaneously , for some graph G ) all of the f i ’s, via an excursioninto commutative algebra.We begin with some definitions. Fix a field k . Let A be a k -graded alge-bra , that is, A is a commutative ring containing k as a subring, that can bewritten as a vector space direct sum A = (cid:77) d ≥ A d , over k , with the property3hat A i A j ⊆ A i + j for all i and j (we call elements in some A i homogeneous ,and A i is called the d -th graded component of A ). The graded k -algebra A is standard if it is generated (as a ring) by a finite set of elements in A . Ourprototypical example of a standard k -graded algebra is the polynomial ring k [ x , x , . . . , x n ] in variables x , x , . . . , x n . Note that for any standard graded k -algebra A = (cid:77) d ≥ A d that is a quotient of a polynomial ring by a homogenousideal, a k -basis for A d is simply the monomials in A d .The Stanley-Reisner complex of an ideal I of a standard graded k -algebra A (generated by x , x , . . . , x n in A ) is the (simplicial) complex whose faces arethe square-free monomials in x , x , . . . , x n not in I (the properties of being anideal ensures that this set is closed under containment). Let I be an ideal of k -algebra A ; I is homogeneous if it is generated by homogeneous elements of A .We write I = (cid:77) d ≥ I d , where I d = A d ∩ I is the d -th graded component of I (it isa k -subspace of I ). For a homogeneous ideal I of A , a square-free monomial M of degree d in x , . . . , x n belongs to exactly one of I d and the Stanley-Reisnercomplex of I (where we identify a face of the complex with the product of itselements). As the total number of monomial of Q of degree d is fixed, we seethat maximizing the number of faces of size d in the Stanley-Reisner complexof I corresponds to minimizing the number of monomials of degree d in I .The Hilbert function of the homogeneous ideal I is the function H I : N → N , where H I ( d ) = dim k I d . We call I Gotzmann (see [11]) if for all otherhomogeneous ideals J of A and all d ≥
0, if H I ( d ) = H J ( d ) then H I ( d + 1) ≤ H J ( d + 1). For an ideal of Q which is Gotzmann, its Hilbert function is smallestfor each value of d ∈ N .We will now focus in on a standard graded k -algebra related to independencein graphs. Fix n ≥
1. The
Kruskal-Katona ring , Q = k [ x , . . . , x n ] / (cid:104) x , . . . , x n (cid:105) ,is generated by the square-free monomials; it is clearly a standard graded k -algebra. Let G be a graph on vertices { x , x , . . . x n } . The edge ideal I G is theideal of Q generated by { x i x j | x i x j ∈ E ( G ) } . If a (square-free) monomial of Q is not in I G then that set of vertices cannot contain an edge in G , so it is anindependent set (and vice versa). This means that the Stanely-Reisner complexof our edge ideal I G in the Kruskal-Katona ring is precisely the independencecomplex ∆ ( G ) of our graph G . If we can show that the edge ideal I G isGotzmann in Q , then this means that for each d ≥ I G contains the fewestmonomials of degree d for all d compared to any other such edge ideal. Henceby a previous observation, the f -vector of the independence complex of G willhave the largest entries component-wise compared to any other graph of order n and size m . Thus our graph will have an independence polynomial that isoptimally-greatest.Let I be an ideal in a standard graded k -algebra A , generated by x , . . . , x n ∈ A . Then I is a lexicographic ideal if for any monomials u and v in x , . . . , x n ,whenever v ∈ I and u is lexicographically bigger than v , we have u ∈ I aswell. It is known that lexicographic ideals are Gotzmann in the Kruskal-Katona4ings [13]. As the edges of our family of graphs G n,m, (cid:22) are added in lexicographicorder, it follows that the edge ideal of I G in Q is lexicographic, and henceGotzmann. Therefore the f -vector is maximized, given f , f , f , that is, given, n and m , and so G n,m, (cid:22) is optimally-greatest. We now turn our attention to the existence of optimally-least graphs for theindependence polynomial, and we find here that we can only prove the existenceof optimally-least graphs for certain values of m = m ( n ) (and we do not knowif there are values of n and m for which optimally-least graphs do not exist).We begin with dense graphs. It is obvious that for a graph G of order n and size m (which has (cid:0) n (cid:1) − m many independent sets of cardinality 2) that theindependence polynomial of such a graph G is, for x ≥
0, at least1 + nx + (cid:18)(cid:18) n (cid:19) − m (cid:19) x , and by a previous observation, if there is a graph with this independence poly-nomial, then it is the optimally-least.Turan’s famous theorem states (see, for example [1]) that the maximumnumber of edges of a graph with no triangles is (cid:100) n (cid:101)(cid:98) n (cid:99) = (cid:98) n (cid:99) , with equality iff G is a complete bipartite graph with sides of equal or nearly equal cardinality.It follows, by taking complements, that provided m ≥ (cid:18) n (cid:19) − (cid:106) n (cid:107) then the graph formed by adding any m − ( (cid:0) n (cid:1) − (cid:98) n (cid:99) ) edges to the complementof K (cid:100) n (cid:101) , (cid:98) n (cid:99) , namely K (cid:100) n (cid:101) , (cid:98) n (cid:99) = K (cid:100) n (cid:101) ∪ K (cid:98) n (cid:99) is the optimally-least graph.This gives the following result: Theorem 2
For a given n ≥ and m ≥ (cid:0) n (cid:1) − (cid:98) n (cid:99) , the graph with theoptimally-least independence polynomial is formed from K (cid:100) n (cid:101) ∪ K (cid:98) n (cid:99) by addingin any m − ( (cid:0) n (cid:1) − (cid:98) n (cid:99) ) edges. The independence polynomial of such a graph is nx + (cid:18)(cid:18) n (cid:19) − m (cid:19) x . (cid:3) We can extend this result by utilizing a result of Lov´asz and Simonovits [12],which, answering a conjecture of Erd¨os, showed that for 1 ≤ k < n/
2, the ( K -free) graph of order n and size (cid:98) n / (cid:99) with the fewest number of triangles is5he graph formed from K (cid:100) n (cid:101) , (cid:98) n (cid:99) by adding in edges to a largest cell so that notriangles are formed within the cell. (In such a case, the number of trianglesformed in the graph is k (cid:98) n/ (cid:99) .) As well, a theorem of Fisher and Solow [9]states that for a ≥ b ≥
1, the least number of triangles in a K -free graph oforder n = 2 a + b and size m = 2 ab + a occurs in K a,a,b . By taking complements,we derive: Theorem 3
For a given n ≥ and m = (cid:0) n (cid:1) − (cid:100) n (cid:101)(cid:98) n (cid:99) − k , where ≤ k ≤(cid:98) n/ (cid:99) , the graph with the optimally-least independence polynomial is formedfrom K (cid:100) n (cid:101) ∪ K (cid:98) n (cid:99) by deleting any k edges in K (cid:100) n (cid:101) so that no independent setof size is formed in that part. The independence polynomial of such a graphis nx + (cid:18)(cid:18) n (cid:19) − m (cid:19) x + k (cid:98) n/ (cid:99) x . As well, for any for a ≥ b ≥ , the graph with the optimally-least independencepolynomial with order n = 2 a + b and size m = a ( a −
1) + b ( b − / is K a ∪ K b ,which has independence polynomial nx + (cid:18)(cid:18) n (cid:19) − m (cid:19) x + (cid:18) (cid:18) a (cid:19) + (cid:18) b (cid:19)(cid:19) x . (cid:3) To find sparse families of optimally-least graphs we will look at a graph op-eration which can be done to decrease the value of the independence polynomialon [0 , ∞ ). Let H be a graph which consists of an induced subgraph G , con-taining an edge e = vw , and two other vertices y and z that are isolated, andlet H be the graph formed from H by removing edge e and adding in an edgebetween y and z (see Figure 1 – we set G = G − e ). Clearly G and G havethe same number of vertices and edges. We will show that this removal of anedge to form a K can never increase the independence polynomial (on [0 , ∞ )). zy zyw G G v wv Figure 1: Graphs for Lemma 4. H is the graph G ∪ { y, z } (on the left) and H is the graph G ∪ { y, z } (on the right). Lemma 4
For the graphs in Figure 1, we have I ( H , x ) ≤ I ( H , x ) on [0 , ∞ ) . roof : Note that by Equation (1), our deletion-contraction formula for inde-pendence polynomials, we have I ( H , x ) = (1 + 2 x + x ) I ( G , x )= (1 + 2 x + x )( I ( G − v, x ) + xI ( G − N [ v ] , x ))= (1 + 2 x + x )( I ( G − v, x ) + xI ( G − N [ v ] − w, x ))and I ( H , x ) = (1 + 2 x )( I ( G − v, x ) + xI ( G − N [ v ] , x )) . We also find that I ( G − N [ v ] , x ) ≤ (1 + x ) I ( G − N [ v ] − w, x ) (2)since G − N [ v ] is a subgraph of ( G − N [ v ] − w ) ∪ K .Consider F ( x ) = I ( H , x ) − I ( H , x ). Using our expression for i ( H , x ) and i ( H , x ) and the inequality (2), we can see that F ( x ) = (1 + 2 x ) I ( G − v, x ) + x I ( G − v, x ) + x (1 + 2 x + x ) I ( G − N [ v ] − w, x ) − (1 + 2 x ) I ( G − v, x ) − x (1 + 2 x ) I ( G − N [ v ] , x )= x I ( G − v, x ) + x (1 + x ) I ( G − N [ v ] − w, x )+ x (1 + x ) I ( G − N [ v ] − w, x ) − x (1 + 2 x ) I ( G − [ v ] , x ) ≥ x I ( G − v, x ) + xI ( G − N [ v ] , x )+ x I ( G − N [ v ] , x ) − xI ( G − [ v ] , x ) − x I ( G − N [ v ] , x )= x I ( G − v, x ) − x I ( G − N [ v ] , x ) . Since G − N [ v ] is a subgraph of G − v , we have I ( G − v, x ) ≥ I ( G − N [ v ] , x ).It follows that F ( x ) ≥
0, so I ( H , x ) ≥ I ( H , x ). (cid:3)(cid:3) It follows that if m ≤ n/
2, by pulling out K ’s, we derive: Theorem 5
For a given n ≥ and m ≤ n , the optimally-least graph for theindependence polynomial for x ≥ is mK ∪ ( n − m ) K . (cid:3) k -Independence Polynomials We now look at the optimality of k -independence polynomials and find thesituation is much different for k ≥ k = 2. We will show that incontrast, to k = 2, for all k ≥
3, there does not always exist optimally-greatestnor optimally-least graphs for the k -independence polynomial.Before we do so, we make the following observation, which shows that thenonexistence of optimal graphs can sometimes be derived by considering onlycertain coefficients of the polynomials. 7 bservation 6 Suppose that G and H be graphs on n vertices and m edgesand let k ≥ , with I k ( G, x ) = n (cid:88) j =0 i j ( G ) x j and I k ( H, x ) = n (cid:88) j =0 i j ( H ) x j Then • if i j ( G ) = i j ( H ) for j < (cid:96) but i (cid:96) ( G ) > i (cid:96) ( H ) , then I k ( G, x ) > I k ( H, x ) for x arbitrarily small and • if i j ( G ) = i j ( H ) for t > j but i t ( G ) > i t ( H ) , then I k ( G, x ) > I k ( H, x ) for x arbitrarily large. Note that i j = (cid:0) nj (cid:1) for j < k . For a graph G , let r G = r kG denote the largestvalue of j such that there exists an induced subgraph of G of order j that doesnot contains a k -clique, that is, r G is the largest value of j such that i j ( G ) > k -independence polynomials ( k ≥
3) there does notalways exist optimally-greatest graphs, we will show that for some n and m = m ( n ), there is a unique graph G ∈ S n,m with the largest value of r G in S n,m , thusoptimally-greatest for sufficiently large values of x , but there is another graph H ∈ S n,m with more k -independent sets than G , and hence optimally-greatestthe k -independence polynomial for arbitrarily small values of x ≥ Theorem 7
For k ≥ and l ≥ , and any n > ( k − l ( l − , there does notexist an optimally-greatest graph for the k -independence polynomial of order n and size m = (cid:0) n (cid:1) − ( k − (cid:0) l (cid:1) . Proof : We recall an old well known result by Turan (see [5], for example) thatstates that the unique graph T n,k of order n with the maximum number m n,k of edges in a graph of order n without a k -clique is the complete ( k − (cid:98) n/ ( k − (cid:99) or (cid:100) n/ ( k − (cid:101) . For fixed n > ( k − l ( l − m = (cid:0) n (cid:1) − ( k − (cid:0) l (cid:1) and consider the class S n,m . We define the graph G = G ( n, m ) as the join T ( k − l,m + K n − ( k − l of the Turan graph T ( k − l,m with K n − ( k − l , that is, G is formed from the disjoint union of T ( k − l,k with K n − ( k − l by adding in all edges between them (equivalently, the complementof G , G , consists of k − l , together with n − ( k − l isolatedvertices). A quick calculation shows that G has m edges, and hence belongs to S n,m .We claim first that among all graphs F (cid:48) in S n,m , G has the largest value of r F (cid:48) . Note that r G = ( k − l as the Turan graph T ( k − l,k has no k -clique. Nowif there is a graph F in S n,m with r F > r G , then F has an induced subgraph S on say s > ( k − l vertices with no k -clique, and hence S has at most as manyedges as T s,m . However, then F has at least as many edges as T s,m , which is8trictly more than the number of edges in T ( k − l,m (to see this, think of thecomplements of Turan graphs being the disjoint unions of cliques, and observethat for t > ( k − l , one can form T t,m from T ( k − l,m by successively addingvertices to the cliques to keep them as nearly equal as possible). Thus F wouldcontain more edges than G , the disjoint union of T ( k − l,m and isolated vertices,and hence F would fewer edges than G , a contradiction as both F and G havethe same number of vertices and edges. We conclude no such F exists, so G hasthe maximal r F value. Moreover, by the argument given, if G (cid:48) were a graph in S n,m with r G (cid:48) = r G , then if S is any induced subgraph of G (cid:48) of size r G = ( k − l that has no k -clique, S must have precisely as many edges as T ( k − l,k , which isthe number of edges of G . We conclude that, from Turan’s Theorem, that G (cid:48) isisomorphic to G , which is therefore the unique graph in S n,m with the largest r F -value, and hence the unique graph optimally-greatest for the k -independencepolynomial for x sufficiently large.So if there is an optimally-greatest graph the k -independence polynomial for S n,m , it must be G . We note that if a graph of order n and size m has a minimumnumber of k -cliques, then it has a maximum number of k -independent sets oforder k . We will show now that there is another graph H ∈ S n,m with fewer k -cliques than G , and hence more k -independent sets of size k , and so G cannotbe optimally-greatest the k -independence polynomial as I k ( H, x ) > I k ( G, x ) for x > H be the graph of order n such that H is the disjoint union of ( k − (cid:0) l (cid:1) K ’s and isolated vertices (as n > ( k − l ( l −
1) = 2( k − (cid:0) l (cid:1) , we can find sucha graph of order n ). We can think of H as splitting the edges of the l -cliques of G into edges. Now it is easy to see that for graphs G and G having k ,i and k ,i cliques of order i , respectively, then for any positive integer t , the numberof k -cliques of order t in G + G is (cid:88) i + j = t k ,i k ,j . It follows that it suffices to show that the following graph G = K l + l ( l − K has, for all i , at least as many i -cliques as the graph G = (cid:0) l (cid:1) K , and thatfor i ≥ i -cliques than the latter. For i = 1 and i = 2 they have the same number of i -cliques (as they have the same number ofvertices and edges). For i ≥
2, the number of i -cliques in G is (cid:18) l ( l − i (cid:19) + l (cid:18) l ( l − i − (cid:19) while G has (cid:18)(cid:0) l (cid:1) i (cid:19) i many i -cliques. We set f l,i = (cid:0) l ( l − i (cid:1) + l (cid:0) l ( l − i − (cid:1)(cid:0) ( l ) i (cid:1) i . f l, = 1 as both G and G has the same number of edges. We’ll showthat the sequence is strictly increasing, and hence for i ≥
3, always greater than1, so that G has strictly more i -cliques than G , concluding the proof.Now via some straightforward but tedious calculations, we find that for i ≥ f l,i +1 f l,i = ( l ( l − − i + l ( i + 1))( l ( l − − i + 1)( l ( l − − i )( l ( l − − i + 1 + li ) . It follows that f l,i +1 f l,i > l ( l − − i + l ( i + 1))( l ( l − − i + 1) > ( l ( l − − i )( l ( l − − i + 1 + li ) , which holds iff i ( l − i − > . The latter is true as i, l ≥ f l,i > i ≥
2. Thus H has fewer k -cliquesthan G , hence more k -independent sets of order k and so an optimally-greatestgraph the k -independence polynomial does not exist. (cid:3)(cid:3) We turn finally to the issue of optimally-least k -independence polynomials.From Observation 6, we have that if an optimally-least graph the k -independencepolynomial exists, then it must have the least number of k -independent sets oforder k (or equivalently, for the optimally-least graph the k -independence poly-nomial has the maximum number of k -cliques), since a graph with the latter willbe optimally-least the k -independence polynomial for sufficiently small valuesof x ≥
0. In [5] it was shown that for a graph on n vertices and m = (cid:0) d (cid:1) + r edges (0 ≤ r < d ) and for k ≥
3, the maximum number of cliques of size k is (cid:0) dk (cid:1) + (cid:0) rk − (cid:1) , and a graph which achieves such bounds consists of a K d , a vertex x with N ( x ) ⊆ V ( K d ), and n − d − r < k −
1, since the addition of fewerthan k − k -clique, but for values of r ≥ k − k -cliques, the edges will have to be added to the same vertex.) Such graphsare candidates for optimally-least graphs the k -independence polynomial, butwe will show that for k ≥
3, that optimally-least graphs the k -independencepolynomial do not always exist. Theorem 8
For k ≥ , n ≥ (cid:0) k (cid:1) + k + 1 vertices and m = (cid:0) ( k ) +22 (cid:1) − optimally-least graphs the k -independence polynomial do not exist. k -independence polynomialnear 0. Proof : Let k ≥
3. Let G = (cid:16) K ( k ) +2 − e (cid:17) ∪ ( n − (cid:0) k (cid:1) − K , for some edge e of K ( k ) +2 , and let H = K ( k ) +1 ∪ K k ∪ ( n − (cid:0) k (cid:1) − − k ) K . We know that G is optimally-least for the k -independence polynomial, when x is sufficientlyclose to 0, and H is not. We will now show that H is optimally-least for the k -independence polynomial for arbitrarily large values of x .In G , the largest size of a vertex set that does not contain a K k is n − (cid:0) k (cid:1) + k −
2, taking any of the n − (cid:0) k (cid:1) − e and k − K ( k ) +2 . In H , the largest size of a vertex set that does notcontain a K k is size n − (cid:0) k (cid:1) + k −
3, taking any of the isolated n − (cid:0) k (cid:1) − − k vertices and k − H is optimally-least the k -independence polynomial for arbitrarily large valuesof x , and G is not, so no optimally-least graphs exist for the k -independencepolynomial for these n and m . (cid:3)(cid:3) While we have seen that optimally-greatest graphs exist for independence poly-nomials, we have only be able to prove the existence of optimally-least poly-nomials for a restricted collection of n and m . Our belief is that such graphsalways exist, but there does not seem to be any reasonable family to put forwardas extremal.In terms of optimality for k -independence polynomials, we have seen thatfor all k ≥
3, there are infinitely many values of n and m such that optimally-greatest graphs do not exists, and similarly for optimally-least graphs.11 full characterization of when optimal graphs for the k -independence poly-nomial exist (for k ≥
3, and even for optimally-least for k = 2) remains open. Acknowledgements
J.I. Brown acknowledges support from NSERC (grant application RGPIN 170450-2013). D. Cox acknowledges research support from NSERC (grant applicationRGPIN 2017-04401) and Mount Saint Vincent University.
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