[Paper Review] On the largest real root of independence polynomials of graphs, an ordering on graphs, and starlike trees

Mohammad Reza Oboudi|arXiv (Cornell University)|Mar 13, 2013

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper introduces a novel ordering on graphs based on the dominance of independence polynomials over the interval [ξ(G), 0], where ξ(G) is the largest real root of the independence polynomial. It proves that for all trees of order n, the star Sₙ is the maximum and the path Pₙ is the minimum under this ordering, and establishes a complete characterization of the ordering for starlike trees using degree sequence dominance.

ABSTRACT

Let $G$ be a simple graph of order $n$. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $I(G,x)=\sum_{k=0}^{n} s(G,k) x^{k}$, where $s(G,k)$ is the number of independent sets of $G$ of size $k$ and $s(G,0)=1$. Clearly all real roots of $I(G,x)$ are negative. Let $ξ(G)$ be the largest real root of $I(G,x)$. Let $H$ be a simple graph. By $G \succeq H$ we mean that $I(H,x)\geq I(G,x)$ for every $x$ in the interval $[ξ(G),0]$. We note that $G\succeq H$ implies that $ξ(G)\geq ξ(H)$. Also we let $G\succ H$ if and only if $G\succeq H$ and $I(G,x) eq I(H,x)$. We prove that for every tree $T$ of order $n$, $S_n\succeq T\succeq P_n$, where $S_n$ and $P_n$ are the star and the path of order n, respectively. By $T=T(n_1,\ldots,n_k)$ we mean a tree $T$ which has a vertex $v$ of degree $k$ such that $T\setminus v=P_{n_1-1}+\cdots+P_{n_k-1}$, that is $T\setminus v$ is the disjoint union of the paths $P_{n_1-1},\ldots,P_{n_k-1}$. Let $X=(x_1,\ldots,x_k)$ and $Y=(y_1,\ldots,y_k)$, where $x_1\geq \cdots\geq x_k$ and $y_1\geq\cdots\geq y_k$ are real. By $X\succ Y$, we mean $x_1=y_1,\ldots,x_{t-1}=y_{t-1}$ and $x_t>y_t$ for some $t\in\{1,\ldots,k\}$. We let $X\succ_{d}Y$, if $X eq Y$ and for every $j$, $1\leq j\leq k$, $\sum_{i=1}^{j}x_i\geq \sum_{i=1}^{j}y_i$. Among all trees with fixed number of vertices, we show that if $(m_1,\ldots,m_k)\succ_{d}(n_1,\ldots,n_k)$, then $T(n_1,\ldots,n_k)\succ T(m_1,\ldots,m_k)$. We conjecture that $T(n_1,\ldots,n_k)\succ T(m_1,\ldots,m_k)$ if and only if $(m_1,\ldots,m_k)\succ (n_1,\ldots,n_k)$, where $\sum_{i=1}^kn_i=\sum_{i=1}^km_i$.

Motivation & Objective

• To define and analyze a new partial order on graphs based on the dominance of independence polynomials over the interval [ξ(G), 0].
• To characterize the extremal trees (maximum and minimum) under this ordering among all trees of a given order.
• To investigate the relationship between the degree sequence dominance and the new graph ordering, particularly for starlike trees.
• To determine whether the independence polynomial uniquely determines a starlike tree and whether degree sequence dominance implies the new graph ordering.

Proposed method

• Define ξ(G) as the largest real root of the independence polynomial I(G,x), which is always negative.
• Introduce the relation G ⪰ H if I(H,x) ≥ I(G,x) for all x ∈ [ξ(G), 0], and G ≻ H if G ⪰ H and I(G,x) ≠ I(H,x).
• Use structural decomposition: for starlike trees T(n₁,…,nₖ), remove the central vertex to obtain disjoint paths P_{nᵢ−1}, and analyze the resulting subgraphs.
• Apply induction and edge/vertex deletion techniques to compare independence polynomials of trees with similar structures.
• Define degree sequence dominance ≻_d: (m₁,…,mₖ) ≻_d (n₁,…,nₖ) if partial sums of sorted sequences satisfy ∑_{i=1}^j mᵢ ≥ ∑_{i=1}^j nᵢ for all j, with strict inequality at some j.
• Use known results on independence polynomial behavior under graph operations (e.g., edge deletion) to compare polynomial values and derive the ordering.

Experimental results

Research questions

• RQ1 Does the degree sequence dominance (≻_d) imply the graph ordering ≻ for starlike trees?
• RQ2 Is the independence polynomial of a starlike tree uniquely determined by its degree sequence?
• RQ3 Are the star Sₙ and the path Pₙ the maximum and minimum elements, respectively, under the ≻ ordering among all trees of order n?
• RQ4 Does the graph ordering ≻ imply degree sequence dominance for general trees?
• RQ5 If two trees have identical independence polynomials, must they have the same degree sequence?

Key findings

• For every tree T of order n, Sₙ ≻ T ≻ Pₙ holds, meaning the star is the maximum and the path is the minimum under the ≻ ordering.
• If (m₁,…,mₖ) ≻_d (n₁,…,nₖ), then T(n₁,…,nₖ) ≻ T(m₁,…,mₖ) for starlike trees with the same number of vertices.
• The independence polynomial uniquely determines the degree sequence of a starlike tree: if I(T₁,x) = I(T₂,x), then D_{T₁} = D_{T₂}.
• The conjecture is proposed that T(n₁,…,nₖ) ≻ T(m₁,…,mₖ) if and only if (m₁,…,mₖ) ≻ (n₁,…,nₖ), extending the dominance result.
• The largest real root ξ(G) is non-decreasing under the ≻ ordering: G ≻ H implies ξ(G) ≥ ξ(H).
• For starlike trees, the ordering ≻ is completely characterized by degree sequence dominance ≻_d, with equality in polynomials only when degree sequences are identical.