Optimal Graphs for Independence and k-Independence Polynomials

Abstract

The independence polynomial I(G, x) of a finite graph G is the generating function for the sequence of the number of independent sets of each cardinality. We investigate whether, given a fixed number of vertices and edges, there exists optimally-least (optimally-greatest) graphs, that are least (respectively, greatest) for all non-negative x. Moreover, we broaden our scope to k-independence polynomials, which are generating functions for the k-clique-free subsets of vertices. For $k \ge 3$, the results can be quite different from the $k = 2$ (i.e. independence) case.

Appendix

We provide here a purely algebraic proof to Theorem 1.

We refer the reader [7, 10] for introductions to complexes and their connections to commutative algebra. A simplicial complex $\mathcal {C}$ is an ordered pair (E, I) consisting of a finite ground set E and a collection of subsets, I of E where $\emptyset \in I$ and if $X\in I$ and $Y \subseteq X$ then $Y\in I$. That is, the elements of I are closed under subsets. The elements of $\mathcal {C}$ are called faces and the maximal elements are called facets. The maximum size, d of a face is called the dimension of the complex. The associated F-vector for a simplicial complex $\mathcal {C}$ is the vector $\langle F_0,F_1,\ldots ,F_d \rangle$, where $F_i$ is the number of faces of size i. Let $\sigma _1,\sigma _2,\ldots ,\sigma _r$ be the facets of $\mathcal {C}$. As the faces of $\mathcal {C}$ are closed under subsets, we will represent the complex by its set of facets, that is, $\mathcal {C}=\{\sigma _1,\sigma _2,\ldots ,\sigma _r\}$.

The independence complex of graph G of order n and size m, $\varDelta _{2}(G)$, has as its faces the independent sets of G (these are obviously closed under containment, and hence a complex). The f-vector of $\varDelta _{2}(G)$ is $(1,n,{{n} \atopwithdelims (){2}} - m,f_{3},\ldots ,f_{\beta })$, where $f_{i}$ is the number of faces of cardinality i in the complex (and $\beta$ is the independence number of G, which is the same as the dimension of the complex). We will show that we can maximize the independence polynomial on $[0,\infty )$ by maximizing (simultaneously, for some graph G) all of the $f_{i}$’s, via an excursion into commutative algebra.

We shall need some further definitions. Fix a field ${\mathbf k}$. Let A be a ${\mathbf k}$-graded algebra, that is, A is a commutative ring containing ${\mathbf k}$ as a subring, that can be written as a vector space direct sum $\displaystyle {A = \bigoplus _{d\ge 0} A_{d}}$, over ${\mathbf k}$, with the property that $A_{i}A_{j} \subseteq A_{i+j}$ 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 ${\mathbf k}$-algebra A is standard if it is generated (as a ring) by a finite set of elements in $A_{1}$. Our prototypical example of a standard ${\mathbf k}$-graded algebra is the polynomial ring ${\mathbf k}[x_{1},x_{2},\ldots ,x_{n}]$ in variables $x_{1},x_{2},\ldots ,x_{n}$.

Let I be an ideal of ${\mathbf k}$-algebra A; I is homogeneous if it is generated by homogeneous elements of A. We write $\displaystyle {I = \bigoplus _{d\ge 0} I_{d}}$, where $I_{d} = A_{d} \cap I$ is the d-th graded component of I (it is a ${\mathbf k}$-subspace of I). Note that for any standard graded ${\mathbf k}$-algebra $\displaystyle {A = \bigoplus _{d\ge 0} A_{d}}$ that is a quotient of a polynomial ring by a homogenous ideal, a ${\mathbf k}$-basis for $A_{d}$ is simply the monomials in $A_{d}$.

The Stanley-Reisner complex of an ideal I of a standard graded ${\mathbf k}$-algebra A (generated by $x_{1},x_{2},\ldots ,x_{n}$ in $A_{1}$) is the (simplicial) complex whose faces are the square-free monomials in $x_{1},x_{2},\ldots ,x_{n}$ not in I (the properties of being an ideal ensures that this set is closed under containment). For a homogeneous ideal I of A, a square-free monomial M of degree d in $x_{1},\ldots ,x_{n}$ belongs to exactly one of $I_{d}$ and the Stanley-Reisner complex of I (where we identify a face of the complex with the product of its elements). As the total number of monomial of Q of degree d is fixed, we see that maximizing the number of faces of size d in the Stanley-Reisner complex of I corresponds to minimizing the number of monomials of degree d in I.

The Hilbert function of the homogeneous ideal I is the function $H_{I}:{\mathbb N} \rightarrow {\mathbb N}$, where $H_{I}(d) = \text{ dim }_{\mathbf k}I_d$. We call I Gotzmann (see [13]) if for all other homogeneous ideals J of A and all $d \ge 0$, if $H_{I}(d)=H_{J}(d)$ then $H_{I}(d+1)\le H_{J}(d+1)$. For an ideal of Q which is Gotzmann, its Hilbert function is smallest for each value of $d \in {\mathbb N}$.

We will now focus in on a standard graded ${\mathbf k}$-algebra related to independence in graphs. Fix $n \ge 1$. The Kruskal-Katona ring, $Q = {\mathbf k}[x_{1},\ldots ,x_{n}]/\langle x_{1}^{2},\ldots ,x_{n}^{2}\rangle$, is generated by the square-free monomials; it is clearly a standard graded k-algebra. Let G be a graph on vertices $\{x_1,x_2,\ldots x_n\}$. The edge ideal $I_{G}$ is the ideal of Q generated by $\{x_ix_j \mid x_ix_j \in E(G) \}$. 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 an independent set (and vice versa). This means that the Stanely-Reisner complex of our edge ideal $I_{G}$ in the Kruskal-Katona ring is precisely the independence complex $\varDelta _{2}(G)$ of our graph G. If we can show that the edge ideal $I_{G}$ is Gotzmann in Q, then this means that for each $d \ge 0$, $I_{G}$ contains the fewest monomials of degree d for all d compared to any other such edge ideal. Hence by a previous observation, the f-vector of the independence complex of G will have 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 is optimally-greatest.

Let I be an ideal in a standard graded ${\mathbf k}$-algebra A, generated by $x_{1},\ldots ,x_{n} \in A_{1}$. Then I is a lexicographic ideal if for any monomials u and v in $x_{1},\ldots ,x_{n}$, whenever $v \in I$ and u is lexicographically bigger than v, we have $u \in I$ as well.

It is known that lexicographic ideals are Gotzmann in the Kruskal-Katona rings [19]. As the edges of our family of graphs $G_{n,m,\preceq }$ are added in lexicographic order, it follows that the edge ideal of $I_G$ in Q is lexicographic, and hence Gotzmann. Therefore the f-vector is maximized, given $f_0, f_1, f_2$, that is, given, n and m, and so $G_{n,m,\preceq }$ is optimally-greatest.