On graphs whose flow polynomials have real roots only

Let $G$ be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the flow polynomial $F(G,\lambda)$ of $G$ are integers if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether a bridgeless graph $G$ for which $F(G,\lambda)$ has real roots only must be the dual of some chordal and plane graph. We conclude that the answer of this problem for $G$ is positive if and only if $F(G,\lambda)$ does not have any real root in the interval $(1,2)$. We also prove that for any non-separable and $3$-edge connected $G$, if $G-e$ is also non-separable for each edge $e$ in $G$ and every $3$-edge-cut of $G$ consists of edges incident with some vertex of $G$, then all roots of $P(G,\lambda)$ are real if and only if either $G\in \{L,Z_3,K_4\}$ or $F(G,\lambda)$ contains at least $9$ real roots in the interval $(1,2)$, where $L$ is the graph with one vertex and one loop and $Z_3$ is the graph with two vertices and three parallel edges joining these two vertices.


Introduction
The graphs considered in this paper are undirected and finite, and may have loops and parallel edges. For any graph G, let V (G), E(G), P (G, λ) and F (G, λ) be the set of vertices, the set of edges, the chromatic polynomial and the flow polynomial of G. The roots of P (G, λ) and F (G, λ) are called the chromatic roots and the flow roots of G respectively. As P (G, λ) = 0 (resp. F (G, λ) = 0) whenever G contains loops (resp. bridges), we will assume that G is loopless (resp. bridgeless) when P (G, λ) (resp. F (G, λ)) is considered.
The chromatic polynomial P (G, λ) of G is a function which counts the number of proper λ-colourings whenever λ is a positive integer. A chordal graph G is a graph in which every subgraph of G induced by a subset of V (G) is not isomorphic to any cycle of length larger than 3. It is known that if G is chordal, then all chromatic roots of G are non-negative integers (see [6,16,14]). Some non-chordal graphs also have this property (see [2,6,7,5,15]). Meanwhile, there are graphs which have real chromatic roots only but also have non-integral chromatic roots. For example, when s ≥ 7, the graph H s obtained from K s by subdividing a particular edge once is such a graph, as P (H s , λ) = λ(λ − 1) · · · (λ − s + 2)(λ 2 − sλ + 2s − 3). (1.1) However, it is still unknown if there is a planar graph G with this property, i.e., G has real chromatic roots only but also contains non-integral chromatic roots. Due to Tutte [20], P (G, λ) = λF (G * , λ) holds for any connected plane graph G, where G * is the dual of G. Thus, equivalently, it is unknown if there is a planar graph G which has real flow roots only but also has non-integral flow roots. Actually it is also unknown if there is a non-planar graph with this property. It is natural to consider the following problem.

Problem 1 Is there a bridgeless graph which has real flow roots only but also contains nonintegral flow roots?
By the following result due to Kung and Royle [13], Problem 1 is equivalent to whether there exists a graph G which is not the dual of any plane and chordal graph but has real flow roots only. If there does not exist any graph asked in Problem 1, then every graph with real flow roots only must be the dual of some chordal and plane graph.

Theorem 1 ([13]) If G is a bridgeless graph, then its flow roots are integral if and only if
G is the dual of a chordal and plane graph.
In this paper, let R be the family of bridgeless graphs which have real flow roots only. We will focus on graphs in R and mainly show that for any graph G ∈ R, all flow roots of G are integers if and only if G does not contain any real flow roots in the interval (1, 2).
A vertex x in a connected G is called a cut-vertex if G − x has more components than G has, where G − x is the graph obtained from G by deleting x and all edges incident with x. A graph G = (V, E) is said to be non-separable if either |E| = |V | = 1 or G is connected without loops or cut-vertices. An edge-cut S of a graph G = (V, E) is the set of edges joining verteces in V 1 to vertices in V 2 for some partition The definition of the flow polynomial of a graph G is given in (2.2). By the second equality in (2.2), F (G, λ) = 0 holds if G contains bridges. By the second and the fifth equalities in (2.2), F (G, λ) = F (G/e, λ) if e is one edge in a 2-edge-cut of G. For this reason, the study of flow polynomials can be restricted to 3-edge connected graphs. By Lemmas 1, 2 and 3, the flow polynomial of any graph can be expressed as the product of flow polynomials of graphs G satisfying the following conditions, divided by (λ − 1) a (λ − 2) b for some non-negative integers a or b: (i) G is non-separable and 3-edge connected; (ii) G does not contain any proper 3-edge-cut; and (iii) G − e is non-separable for each edge e in G.
Let R 0 be the family of those graphs in R which satisfying conditions (i), (ii) and (iii) above. By Lemmas 1, 2 and 3, there exists a graph asked in Problem 1 belonging to R if and only if there exists a graph asked in Problem 1 belonging to R 0 . Thus the study of Problem 1 can be focused on graphs in R 0 .
Let W (G) be the set of vertices in a graph G of degrees larger than 3 and letd(G) be the mean of degrees of vertices in W (G). Let L denote the graph with one vertex and one loop and let Z k denote the graph with two vertices and k parallel edges joining these two vertices.
Our main result in this paper is the following one.
Theorem 2 Assume that G = (V, E) is any graph in R. Remark: Theorem 2 (ii) implies that Theorem 1 holds for all graphs in R 0 .
Interestingly, Theorems 1 and 2 imply three equivalent statements on a bridgeless graph which has real flow roots only.

Basic results on flow polynomials
Let G = (V, E) be a finite graph with vertex set V and edge set E and let D be an orientation of G. For any finite additive Abelian group Γ, a Γ-flow on D is a mapping φ : E → Γ such that is the set of loopless arcs in D with tail v (resp. with head v). If φ(e) = 0 for all e ∈ E, then a Γ-flow φ on D is called a nowhere-zero Γ-flow on D. For any integer q ≥ 2, a nowhere-zero q-flow of G is defined to be a nowhere-zero Z-flow ψ such that |ψ(e)| ≤ q − 1 for all e ∈ E, where Z is the additive group consisting of all integers. Tutte [21] showed that G has a nowhere-zero q-flow if and only if it has a nowhere-zero Γ-flow, where q is the order of Γ.
The flow polynomial F (G, λ) of a graph G is a function in λ which counts the number of nowhere-zero Γ-flows on D whenever λ is equal to the order of Γ. Note that the definition of F (G, λ) does not depend on the selection of D and the additive Abelian group Γ but on G and the order of Γ. The function F (G, λ) can also be obtained recursively by the following rules (see Tutte [22]): if e is not a loop nor a bridge, where G/e and G − e are the graphs obtained from G by contracting e and deleting e respectively and G 1 ∪ G 2 is the disjoint union of graphs G 1 and G 2 .
A block of G is a maximal subgraph of G with the property that it is non-separable. By (2.2), the following result can be obtained.
If G is non-separable, F (G, λ) can also be factorized when G − e is separable for some edge e or G has a proper 3-edge-cut S. The results have been given in [11] (see [4,10,12] also).

Lemma 2 ([11])
Let G be a bridgeless connected graph, v be a vertex of G, e = u 1 u 2 be an edge of G, and H 1 and , as shown in Figure 1. Then If G has an edge-cut S with 2 ≤ |S| ≤ 3, then F (G, λ) also has a factorization [11].

Lemma 3 ([11])
Let G be a bridgeless connected graph, S be an edge-cut of G and H 1 and H 2 be the sides of S, as shown in Figure 2 It is not difficult to prove that for any bridgeless graph G, F (G, λ) has no zero in (−∞, 1). But 1 is a zero of F (G, λ) whenever G is not an empty graph. These conclusions can be obtained by equalities in (2.2). The next zero-free interval for flow polynomials is (1, 32/27], due to Wakelin [17]. For any integer k ≥ 0, let ξ k be the supremum in (1,2] such that F (G, λ) is non-zero in the interval (1, ξ k ) for all bridgeless graphs G with at most k vertices of degrees larger than 3 (i.e., |W (G)| ≤ k). Clearly that ξ 0 ≥ ξ 1 ≥ ξ 2 ≥ · · ·. It is shown in [4] that each ξ k can be determined by the flow roots of graphs from a finite set.

Proof.
By Theorem 4, ξ k is determined by the flow roots of graphs from a finite set Θ k . Thus ξ k is the flow root of some graph in Θ k . By Theorem 3, ξ k > 32/27. ✷

Graphs with real flow roots only
In this section, we assume that G = (V, E) is a connected and bridgeless graph and r, γ, α, k, R and ω are some invariants related to G defined below: where v i is the number of vertices in G of degree i; (iii) γ is the number of 3-edge-cuts of G; (v) R is the multiset of real roots of F (G, λ) in (1, 2); and If we take another graph H, the above parameters related to H are denoted by r(H), α(H), γ(H), k(H), R(H) and ω(H) respectively. It is straightforward to verify the following relations on these parameters.
It can be verified by (2.2) that F (G, λ) is a polynomial of order r. Furthermore, if G is 3-edge connected, the coefficients of the three leading terms can be expressed in terms of r, |E| and γ (see [13]).
Recall that L is the graph with one vertex and one loop and R is the family of bridgeless graphs which have real flow roots only. Obviously, we have the following conclusion on r.
From now on, we assume that G is a 3-edge connected graph in R. By Lemma 5, we can get a lower bound for γ in terms of |E| and r. Proof. (i) It is known that any non-empty graph does not have nowhere-zero 1-flows, i.e., F (G, λ) has a root 1. Write By Lemma 5, a 1 + 1 = |E| and a 2 + a 1 = |E| 2 − γ. So γ = |E| 2 − a 2 − |E| + 1. Since all roots of F (G, λ) are real, applying Lemma 3.1 in [13] or the Newton Inequality [8] to the coefficients of the three leading terms in the second factor of the right-hand side of (3.1), we have

2)
1 It is known that G has a nowhere-zero 2-flow if and only if every vertex of G has an even degree.
where the inequality is strict if (|E| − 1)/(r − 1) is not a root of F (G, λ). Note that if (|E| − 1)/(r − 1) is not an integer, it is not a root of F (G, λ). Thus and (i) follows.
(ii). It can be obtained similarly. As G is not even, both 1 and 2 are flow roots of G. Write Applying the idea used in the proof of (i), we have c 1 = |E| − 3, |E| 2 − γ = c 2 + 3c 1 + 2 and where the inequality is strict whenever |E|−3 r−2 is not an integer. Thus where the inequality is strict if F (G, λ) has some real roots in (2, ∞); (vii) k < 11 27 |V | + b+9 54 . Proof.
(iv). By (ii) and the definition ofd(G), implying thatd (v). Let t = |R(G)|, i.e., t is the number of real roots of F (G, λ) in the interval (1,2). Thus t is the sum of the multiplicities of all flow roots of G in (1, 2).
By Theorem 3, one root of F (G, λ) is 1 with multiplicity b, exactly t of its roots are in (1,2) and (r − t − b) of its roots are at least 2. As |E| is the sum of all flow roots of G, we have (3.9) implying that ω ≥ |E| − 2|V | + 1 − b as r = |E| − |V | + 1, where the inequality is strict if F (G, λ) has some real roots in (2, ∞).
On the other hand, where the last inequality is from (v). So (3.10) and (3.11) imply that Then it follows that where the last inequality follows from the fact that ξ k > 32/27 by Corollary 2.
(vii). By (ii) and (vi), we have Solving this inequality gives that k < 11 27 |V | + b+9 54 . So (vii) holds. ✷ We are now going to establish the following important result. Recall that R 0 is the family of non-separable and 3-edge connected graphs G in R such that G does not contain any proper 3-edge-cut and G − e is non-separable for each edge e in G.
Proof. Suppose that G ∈ {L, Z 3 , K 4 }. Clearly, |V | ≥ 2, as |V | = 1 and G ∈ R 0 imply that G = L. It is easy to verify that all flow roots of Z s are real if and only if s = 3. As G = Z 3 , |V | = 2. Hence |V | ≥ 3.

Proof.
Let Z be the set of graphs in R which contain flow roots not in the set {1, 2, 3}. Suppose that the result fails and G is a graph in Z with the minimum value of |E(G)| such that |E| < |V | + 17 or |R| < 9. We first prove the following claims.
It is easy to verify that for any non-separable graph H of order at most 2, if all flow roots of H are real, then each flow root of G is in {1, 2, 3}. As G ∈ Z, this claim holds.
Claim 5: G − e is non-separable for each edge e in G.
Suppose that G − e is separable for some edge e = u 1 u 2 as shown in Figure 1. By Lemma 2, where G 1 and G 2 are the graphs stated in Lemma 2. Then this claim can be proved similarly as the previous claim.

Proof.
Assume that H is a connected plane graph and H * is its dual. By the equality P (H, λ) = λF (H * , λ) due to Tutte [20], the given conditions implies that H * has real flow roots only. As H is not chordal, P (H, λ) has non-integral roots by the result in [5] that planar graphs with integral chromatic roots are chordal. Thus H * has real flow roots only but also contains non-integral flow roots. By Theorem 6, H * has at least 9 flow roots in (1,2), implying that H has at least 9 chromatic roots in (1,2). Notice that H * has m edges and n faces. By Euler's polyhedron formula, the order of H * is |V (H * )| = m − n + 2. Thus 32n/27 − 5/9 < m ≤ 2n − 8. ✷ We end this article with the following remark.
Remark: By Lemmas 1, 2 and 3, the study of Problem 1 can be restricted to those graphs in the family R 0 . Thus, by Theorem 5, there exist graphs asked in Problem 1 if and only if R 0 − {L, Z 3 , K 4 } = ∅. By Theorem 5 again, for any G ∈ R 0 − {L, Z 3 , K 4 }, G contains at least at least ⌈ 27k 11 − 27 22 ⌉ + 2µ(6 − k) ≥ 9 flow roots in the interval (1, 2), where k = |W (G)| ≥ 3. However, as I know, no much research is conducted on counting the number of real flow roots of a graph in the interval (1, 2), except some study which confirms certain families of graphs having no real flow roots in the interval (1, 2) (see [3,4,10,11,12]