Dong, F. M.

This note presents two results on real zeros of chromatic polynomials. The first result states that if G is a graph containing a q-tree as a spanning subgraph, then the chromatic polynomial P(G, λ) of G has no non-integer zeros in the interval (0, q). Sokal conjectured that for any graph G and any real λ > Δ(G), P(G, λ) > 0. Our second result confirms that it is true if Δ(G) ≥ [n/3] − 1, where n is the order of G.

For any positive integers a, b, c, d, let Ra,b,c,d be the graph obtained from the complete graphs Ka,Kb,Kc and Kd by adding edges joining every vertex in Ka and Kc to every vertex in Kb and Kd. This paper shows that for arbitrary positive integers a, b, c and d, every root of the chromatic polynomial of Ra,b,c,d is either a real number or a non-real number with its real part equal to (a + b + c + d − 1)/2.

For a κ-tree T, a generalization of a tree, the local mean order of sub-κ-trees of T is the average order of sub-κ-trees of T containing a given κ-clique. The problem whether the maximum local mean order of a tree (i.e., a 1-tree) at a vertex is always taken on at a leaf was asked by Jamison in 1984 and was answered by Wagner and Wang in 2016. Actually, they proved that the maximum local mean order of a tree at a vertex occurs either at a leaf or at a vertex of degree 2. In 2018, Stephens and Oellermann asked a similar problem: for any κ-tree T, does the maximum local mean order of sub-κ-trees containing a given κ-clique occur at a κ-clique that is not a major κ-clique of T? In this paper, we give it an affirmative answer.

For any graph 𝘎, the chromatic polynomial of 𝘎 is the function 𝘗 (𝘎, 𝓂) which counts the number of proper m-colorings of 𝘎 for each positive integer m. The DP color function 𝘗ᴅᴘ(𝘎, 𝓂) of 𝘎, introduced by Kaul and Mudrock in 2019, is a generalization of 𝘗 (𝘎, 𝓂) with 𝘗DP(𝘎, 𝓂)≤ 𝘗 (𝘎, 𝓂) for each positive integer 𝓂. Let 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) (resp. 𝘗ᴅᴘ(𝘎)< 𝘗(𝘎)) denote the property that 𝘗ᴅᴘ(𝘎, 𝓂)= 𝘗(G, 𝓂) (resp. 𝘗ᴅᴘ(G, 𝓂)< 𝘗(G, 𝓂)) holds for sufficiently large integers 𝓂. It is an interesting problem of finding graphs 𝘎 for which 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) (resp. 𝘗ᴅᴘ(𝘎, 𝓂)< 𝘗(𝘎, 𝓂)) holds. Kaul and Mudrock showed that if 𝘎 has an even girth, then 𝘗ᴅᴘ(𝘎)< 𝘗(𝘎) and Mudrock and Thomason recently proved that 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) holds for each graph 𝘎 which has a dominating vertex. We shall generalize their results in this article. For each edge e in 𝘎, let 𝓁(e)=∞ if e is a bridge of 𝘎, and let 𝓁(e) be the length of a shortest cycle in 𝘎 containing e otherwise. We first show that if 𝓁(e) is even for some edge e in 𝘎, then 𝘗ᴅᴘ(𝘎)< 𝘗(𝘎) holds. However, the converse statement of this conclusion fails with infinitely many counterexamples. We then prove that 𝘗ᴅᴘ(𝘎)≈ 𝘗(𝘎) holds for every graph G that contains a spanning tree T such that for each e∈E(G)∖E(T), 𝓁(e) is odd and e is contained in a cycle C of length ℓ(e) with the property that 𝓁(e′)< 𝓁(e) for each e′∈E(C)∖(E(T)∪{e}). Some open problems are proposed in this article.

For a graph G of order n ≥ 2, an ordering (x1, x2, . . . , xn) of the vertices in G is called a double-link ordering of G if x1x2 ∈ E(G) and xi has at least two neighbors in {x1, x2, . . . , xi−1} for all i = 3, 4, . . . , n. This paper shows that certain graphs possessing a kind of double-link ordering have no chromatic zeros in the interval (1, 2). This result implies that all graphs with a 2-tree as a spanning subgraph, certain graphs with a Hamiltonian path, all complete t-partite graphs, where t ≥ 3, and all (v(G) − Δ(G) + 1)-connected graphs G have no chromatic zeros in the interval (1, 2).

Using the theory of electrical network, we first obtain simple formulas for the number of spanning trees of a complete bipartite graph containing a certain matching or a certain tree. Then we compute the effective resistances (i.e., resistance distance in graphs) in the nearly complete bipartite graph G(m,n,p) = K_m,n -pK₂ (p≤ min｛m,n｝which extends a recent result (Ye and Yan, 2019) on the effective resistances in G(n,n,p). As a corollary, we obtain the Kirchhoff index of G(m,n,p) which extends a previous result by Shi and Chen. Using the effective resistances in G(m,n,p), we find a formula for the number of spanning trees of G(m,n,p). In the end, we prove a general result for the number of spanning trees of a complete bipartite graph containing several edges in a certain matching and avoiding others.

For a given graph G, let P (G , A) be the chromatic polynomial of G, where A is considered to be a real number. In this paper, we study the bounds for P (G , A )/P (G , A — 1) and P (G , A )/P (G - x, A), where x is a vertex in G, A > n and n is the number of vertices of G.

A matching M in a multigraph G=(V,E) is said to be uniquely restricted if M is the only perfect matching in the subgraph of G induced by V(M) (i.e., the set of vertices saturated by M). For any fixed vertex χ₀ in G, there is a bijection from the set of spanning trees of G to the set of uniquely restricted matchings of size |V|−1 in S(G)− χ₀, where S(G) is the bipartite graph obtained from G by subdividing each edge in G. Thus the notion “uniquely restricted matchings of a bipartite graph H saturating all vertices in a partite set X” can be viewed as an extension of “spanning trees in a connected graph”. Motivated by this observation, we extend the notion “G-parking functions” of a connected multigraph to “B-parking functions” ƒ:X→{−1,0,1,2,⋯} of a bipartite graph H with a bipartition (X,Y) and find a bijection ψ from the set of uniquely restricted matchings of H to the set of B-parking functions of H. We also show that for any uniquely restricted matching M in H with |M|=|X|, if ƒ=ψ(M), then ∑_x∈X^ƒ(x) is exactly the number of elements y∈Y−V(M) which are not externally B-active with respect to M in H, where the new notion “externally B-active members with respect to M in H” is an extension of “externally active edges with respect to a spanning tree in a connected multigraph”.