#Matching and independence polynomials
This page collects real-rooted graph-polynomial examples. The main methods include deletion/interlacing theorems, the compatible-polynomial viewpoint, stable polynomials, and matching-polynomial technology.
Let \(G = (V,E)\) be any graph. The matching polynomial \(\mu_G(x)\) is defined as \[\mu_G(x) \coloneqq \sum_{\substack{M \subseteq E(G) \\ \text{$M$ matching}}} x^{|M|}\] where \(|M|\) is the number of edges in the matching.
In [Thm. 4.2, HL72], it was shown that \(\mu_G(x)\) is real-rooted. Moreover, \(\mu_G(x)\) interlaces \(\mu_{G'}(x),\) where \(G'\) is any graph obtained from \(G\) by removing a vertex. This is a very powerful theorem, but it follows from the more general Chudnovsky–Seymour theorem on independence polynomials below.
A hypergraph generalization (including multigraphs) is considered by N. Amini in [Ami19].
Example (Laplacian matching polynomial).
Let \(d_G(v)\) denote the degree of \(v\) in \(G.\) The Laplacian matching polynomial is the matching-theoretic Laplacian analogue \[\operatorname{LM}(G,x) \coloneqq \sum_{\substack{M \subseteq E(G) \\ \text{$M$ matching}}} (-1)^{|M|} \prod_{v\in V(G)\setminus V(M)} (x-d_G(v)).\] It is real-rooted with nonnegative roots, see [WWM22]. One way to see why this belongs with matching polynomials is the subdivision identity \[\mathsf{M}(S(G),x)=x^{|E(G)|-|V(G)|}\operatorname{LM}(G,x^2),\] where \(S(G)\) is the subdivision graph and \[\mathsf{M}(H,x) \coloneqq \sum_{\substack{M \subseteq E(H) \\ \text{$M$ matching}}} (-1)^{|M|}x^{|V(H)\setminus V(M)|}\] is the usual Heilmann–Lieb normalization of the matching polynomial.
For connected \(G,\) J.-C. Wan, Y. Wang, and A. Mohammadian also show that \(0\) is a root of \(\operatorname{LM}(G,x)\) if and only if \(G\) is a tree; that the number of distinct positive roots is at least the length of the longest path in \(G\); and that \(\operatorname{LM}(G,x)\) and \(\operatorname{LM}(G-e,x)\) have interlacing roots for each edge \(e\) of \(G\) [WWM22].
Example (Fibonacci tilings via matchings).
Recall that the number of ways to tile the one-row Young diagram \((n)\) with tiles of shape \((1)\) and \((2)\) is the Fibonacci number \(F_{n+1}.\) If we then define \[F_0(x) \coloneqq 1,\qquad F_1(x) \coloneqq 1,\qquad F_n(x) \coloneqq F_{n-1}(x) + x\cdot F_{n-2}(x)\] we see that \([x^k] F_n(x)\) is the number of ways to tile \((n)\) using exactly \(k\) tiles of shape \((2).\)
By the Heilmann–Lieb theorem, we can see that for all \(n,\) \(F_n(x)\) is real-rooted, since we can see that \(\mu_G(x) = F_n(x)\) for the path graph with \(n\) vertices.
Example (Touchard polynomials (blocks in set partitions)).
The Touchard polynomials may be defined via \[T_n(x) \coloneqq \sum_{P \in SP(n)} x^{\mathrm{blocks}(P)} = \sum_{k=1}^n S(n,k) x^k.\] The matching approach: [Lem. 5.1, God17] states that the Stirling number \(S(n,k)\) is equal to the number of size \(n-k\) matchings in the bipartite graph \(G_n\) on vertex set \[\{1,2,\dotsc,n\} \cup \{1',2',\dotsc,n'\}\] where \(i\) has an edge with \(j'\) if and only if \(i \lt j.\)
For example, the graph \(G_4 = \{ \{1,2'\}, \{1,3'\}, \{1,4'\}, \{2,3'\}, \{2,4'\}, \{3,4'\} \}\) has \(7=S(4,4-2)\) matchings of size \(2,\) and \(6=S(4,4-1)\) matchings of size \(1.\) We then have \(x^n T_n(x^{-1}) = \mu_{G_n}(x)\) where \(\mu_{G_n}(x)\) is the matching polynomial of \(G_n.\) Since removal of vertices gives interlacing, \(\mu_{G_{n-1}}(x) \interl \mu_{G_n}(x)\) from which interlacing for Touchard polynomials follows.
The rook placement approach: The matching polynomial \(\mu_{G_n}(x)\) for the bipartite graph above is the rook polynomial of the staircase board of size \(n-1.\)
Example (r-Lah number polynomials and \(r\)-Touchard polynomials).
An interpretation of \(r\)-Lah polynomials and \(r\)-Stirling numbers as matching polynomials is given in [NR21].
#Rook polynomials
The rook polynomial of a board \(B\) is defined as \[R_B(z) \coloneqq \sum_k r_k(B) z^k\] where \(r_k(B)\) is the number of ways to place \(k\) non-attacking rooks on the board. Here, the board can be any subset of some \(n {\times} n\)-board. Note that \(R_B(x)\) is exactly a matching polynomial, \(\mu_G(x),\) for a bipartite graph \(G\) on vertex set \(\{v_1,\dotsc,v_n,v'_1,\dotsc,v'_n\}\) where \((v_i,v'_j)\) is an edge if and only if \((i,j)\) is a cell of the board \(B.\)
A. Nijenhuis [Nij76] proved that the rook polynomials are always real-rooted. In fact, Nijenhuis shows a weighted result involving permanents.
Theorem (Nijenhuis, 1976).
Let \(A = (a_{ij})\) be a non-negative \(m\times n\)-matrix. Then \[R_{A}(z) \coloneqq \sum_{\rho} (-z)^{\length(\rho)} \prod_{j=1}^{\length(\rho)} a(\rho_j)\] where we sum over all ways to place rooks on the \(m\times n\)-board, and \(a(\rho_j)\) is the value of the matrix at position given by the \(j\)th rook.
The hit polynomial of a board can also be defined in terms of the \(r_k(B).\) Let \(B\) be a subset of the \(n\times n\)-board. The hit polynomial is then defined as \[T_B(z) \coloneqq \sum_{k =0}^n k! r_{n-k}(B)(z-1)^{n-k} = \sum_{j=0}^n h_j(B) z^j.\] Here, \(h_j(B)\) is the number of ways to place \(n\) rooks on the \(n \times n\)-board, with exactly \(j\) rooks in \(B,\) see [KR46, Hag00].
A theorem by Haglund–Ono–Wagner [HOW99] states that if \(B\) is a Ferrers board (the same as a Young diagram), then \(T_B(z)\) is real-rooted.
#Independence polynomials for claw-free graphs
Let \(G = (V,E)\) be any graph. The independence polynomial \(I(G,x)\) is defined as \[I(G,x) \coloneqq \sum_{\substack{A \subseteq V(G) \\ \text{$A$ independent}}} x^{|A|}\] where \(|A|\) is the number of vertices in the independent set.
A graph is called claw-free if no induced subgraph is isomorphic to the graph with edges \(\{\{1,2\},\{1,3\},\{1,4\}\}.\)
If \(G\) is any simple graph, its line graph \(L(G)\) is also simple. Moreover, \(L(G)\) is claw-free, and \(I(L(G),x) = \mu_G(x).\)
Theorem (Chudnovsky–Seymour 2007, [CS07]).
Suppose \(G\) is claw-free. Then \(I(G,x)\) is real-rooted. As a corollary, we get the Heilmann–Lieb theorem on matching polynomials.
An alternative proof, closer in spirit to the Heilmann–Lieb proof for matching polynomials, is due to F. Bencs [Ben18]. It uses Christoffel–Darboux type identities for independence polynomials; this route is also useful for proving interlacing statements via Wronskian inequalities.
Proof
This very briefly illustrates the ideas. Note that if \(v \in V(G)\) and \(\operatorname{nbh}(v)\) is the closed neighborhood consisting of \(v\) and all vertices adjacent to \(v,\) then \[I(G,x) = I(G \setminus \{v\} ,x) + x \cdot I(G \setminus \operatorname{nbh}(v),x).\] Essentially, the above sum is broken down into the cases where \(v \notin A\) and where \(v \in A.\) Observe that if \(G\) is claw-free, then any induced subgraph is also claw-free. Chudnovsky–Seymour then use the notion of compatible polynomials to obtain their result.
This result was conjectured by R. Stanley in [Sta98], a vertex-weighted version is given by A. Engström [Eng07] and an alternative (non-vertex-deletion-recursive) proof is given by B. Lass [Las12]. Finally, a multivariate refinement is given by [LR19] where one obtains stable polynomials.
For general graphs, independence polynomials need not be real-rooted. Nevertheless, there are useful partial results.
Theorem (Brown–Dilcher–Nowakowski, [BDN00]).
For any graph \(G,\) a zero of \(I(G,x)\) of smallest modulus is real. Moreover, if \(G\) is well-covered with independence number \(\beta(G),\) then all zeros of \(I(G,x)\) lie in the annulus \[\frac{1}{|V(G)|}\leq |z|\leq \beta(G),\] with equality on the boundary only when \(G\) is complete. Finally, every well-covered graph \(G\) is an induced subgraph of a well-covered graph \(H\) with \(\beta(H)=\beta(G)\) such that \(I(H,x)\) has only simple real zeros.
The last statement shows that real-rooted independence polynomials are plentiful among well-covered graph expansions, even though well-covered graphs themselves may have non-real zeros. For caterpillar trees, there are also broad unimodality results for independence polynomials, see [BES18]; these are unimodality results rather than real-rootedness theorems. R. Pandey studies independence polynomials of generalized Petersen graphs \(\mathrm{GP}(n,k)\) using a transfer-matrix computation [Pan26]. The numerical evidence suggests a parity dichotomy: for even \(k\) the roots appear to be real and negative, while for odd \(k\) they form complex conjugate patterns. The resulting conjecture is that \(I(\mathrm{GP}(n,k),x)\) is real-rooted if and only if \(k\) is even.
Y.-T. Xie and S.-J. Xu study the complementary dependence polynomial \[D(G,x) \coloneqq \sum_k d_k(G)x^k,\] where \(d_k(G)\) is the number of \(k\)-vertex subsets which are not independent [XX24]. They prove ultra-log-concavity for \(D(G,x)\) when \(G\) is \(K_2\cup 2K_1\)-free or has an independent set of size \(|V(G)|-2,\) and they characterize the real-rooted cases. Up to isolated vertices, the real-rooted examples are \(K_2,\) \(P_3,\) \(K_3,\) \(2K_2,\) \(P_4,\) \(C_4,\) and two small exceptional graphs shown in their Fig. 1.
A. Engström proved the following weighted generalization of the Chudnovsky–Seymour theorem:
Theorem (A. Engström, [Thm. 2.5, Eng07]).
Let \(G = (V,E)\) be a graph, and \(w : V \to \setR\) be vertex weights. The weighted independence polynomial \(I(G,w,x)\) is defined as \[I(G,w,x) \coloneqq \sum_{\substack{A \subseteq V(G) \\ \text{$A$ independent}}} x^{|A|} \prod_{v\in A} w(v).\] If \(G\) is claw-free and all weights are non-negative, then \(I(G,w,x)\) is real-rooted.
#Chow polynomials
In [BV25], the authors show various real-rootedness results regarding Chow polynomials. E. Hoster and C. Stump prove real-rootedness of Chow and augmented Chow polynomials for finite graded simplicial posets with a top element whenever the \(h\)-vector is positive [HS25]. This includes Cohen–Macaulay simplicial posets and, in particular, lattices of flats of uniform matroids. They also show that the Chow polynomial of the dual poset interlaces the augmented Chow polynomial.
P. Brändén and L. S. M. Leite study the conjecture of C. A. Athanasiadis and A. Kalampogia-Evangelinou that chain polynomials of geometric lattices have only real zeros [BL25]. They prove it for perfect matroid designs, Dowling lattices, and a class containing lattices of flats of paving matroids, and analyze how the property behaves under several lattice operations.
Theorem (Alexandersson, [Thms. 1.1 and 4.2, Ale26]).
Let \(\operatorname{Smir}_{n,m}\) be the words in \([m]^n\) whose adjacent letters differ. Then \[H_{NC_{n+1}}(t)=\frac{1}{n+1} \sum_{w\in\operatorname{Smir}_{n,n+1}}t^{\des(w)}\] is the Chow polynomial of the noncrossing partition lattice \(NC_{n+1}\) and is real-rooted for every \(n\geq1.\) More strongly, for fixed \(n\) and \(m\geq2,\) the refinements by final letter form an interlacing sequence.
The proof uses the same last-letter recurrence and matrix interlacing criterion as the word-enumerator argument above.
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