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The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. We determine the unique non-starlike non-caterpillar tree with maximal distance spectral radius.

In 2018, Dvořák and Postle introduced a generalization of proper coloring, the so‐called DP‐coloring. For any graph G $G$, the DP‐chromatic number χDP(G) ${\chi }_{DP}(G)$ of G $G$ is defined analogously with the chromatic number χ(G) $\chi (G)$ of G $G$. In this article, we show that χDP(G∨Ks)=χ(G∨Ks) ${\chi }_{DP}(G\vee {K}_{s})=\chi (G\vee {K}_{s})$ holds for s=4(χ(G)+1)|E(G)|2χ(G)+1 $s=\unicode{x02308}\frac{4(\chi (G)+1)|E(G)|}{2\chi (G)+1}\unicode{x02309}$, where G∨Ks $G\vee {K}_{s}$ is the join of G $G$ and a complete graph with s $s$ vertices. As a result, ZDP(n)≤n2−(n+3)∕2 ${Z}_{DP}(n)\le {n}^{2}-(n+3)\unicode{x02215}2$ holds for every integer n≥2 $n\ge 2$, where ZDP(n) ${Z}_{DP}(n)$ is the minimum nonnegative integer s $s$ such that χDP(G∨Ks)=χ(G∨Ks) ${\chi }_{DP}(G\vee {K}_{s})=\chi (G\vee {K}_{s})$ holds for every graph G $G$ with n $n$ vertices. Our result improves the best current upper bound 1.5n2 $1.5{n}^{2}$ of ZDP(n) ${Z}_{DP}(n)$ due to Bernshteyn, Kostochka, and Zhu. [ABSTRACT FROM AUTHOR]

For any connected graph G $G$, let P(G,m) $P(G,m)$ and PDP(G,m) ${P}_{DP}(G,m)$ denote the chromatic polynomial and Dvořák and Postle (DP) color function of G $G$, respectively. It is known that PDP(G,m)≤P(G,m) ${P}_{DP}(G,m)\le P(G,m)$ holds for every positive integer m $m$. Let DP≈ $D{P}_{\approx }$ (resp., DP< $D{P}_{\lt }$) be the set of graphs G $G$ for which there exists an integer M $M$ such that PDP(G,m)=P(G,m) ${P}_{DP}(G,m)=P(G,m)$ (resp., PDP(G,m)

In this article, we extend Moon's classic formula for counting spanning trees in complete graphs containing a fixed spanning forest to complete bipartite graphs. Let (X,Y) $(X,Y)$ be the bipartition of the complete bipartite graph Km,n ${K}_{m,n}$ with ∣X∣=m $| X| =m$ and ∣Y∣=n $| Y| =n$. We prove that for any given spanning forest F $F$ of Km,n ${K}_{m,n}$ with components T1,T2,...,Tk ${T}_{1},{T}_{2},\ldots ,{T}_{k}$, the number of spanning trees in Km,n ${K}_{m,n}$ which contain all edges in F $F$ is equal to 1mn∏i=1k(min+nim)1−∑i=1kminimin+nim, $\frac{1}{mn}\left(\prod _{i=1}^{k}({m}_{i}n+{n}_{i}m)\right)\left(1-\sum _{i=1}^{k}\frac{{m}_{i}{n}_{i}}{{m}_{i}n+{n}_{i}m}\right),$ where mi=∣V(Ti)∩X∣ ${m}_{i}=| V({T}_{i})\cap X| $ and ni=∣V(Ti)∩Y∣ ${n}_{i}=| V({T}_{i})\cap Y| $ for i=1,2,...,k $i=1,2,\ldots ,k$. [ABSTRACT FROM AUTHOR]

The chromatic polynomial P(G,x) of a graph G of order n can be expressed as ∑i=1n(−1)n−iaixi, where ai is interpreted as the number of broken‐cycle‐free spanning subgraphs of G with exactly i components. The parameter ϵ(G)=∑i=1n(n−i)ai∕∑i=1nai is the mean size of a broken‐cycle‐free spanning subgraph of G. In this article, we confirm and strengthen a conjecture proposed by Lundow and Markström in 2006 that ϵ(Tn)<ϵ(G)<ϵ(Kn) holds for any connected graph G of order n which is neither the complete graph Kn nor a tree Tn of order n. The most crucial step of our proof is to obtain the interpretation of all ai's by the number of acyclic orientations of G. [ABSTRACT FROM AUTHOR]

D. Bokal proved that the crossing number is additive for the zip product under the condition of having two coherent bundles in the zipped graphs. This property is very effective when dealing with the crossing numbers of (capped) Cartesian product of trees with graphs containing a dominating vertex. In this paper, we first prove that the crossing number is still additive for the zip product under a weaker condition. Based on the new condition, we then establish some general expressions for bounding the crossing numbers of (capped) Cartesian product of trees with graphs (possibly without dominating vertex). Exact values of the crossing numbers of Cartesian product of trees with most graphs of order at most five are obtained by applying these expressions, which extend some previous results due to M. Klešč. In fact, our results can also be applied to deal with Cartesian product of trees with graphs of order more than five. [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a matching‐covered graph and X be an edge set of G. X is said to be feasible if there exist two perfect matchings M1 and M2 in G such that |M1∩X|≢|M2∩X| (mod 2). For any V0⊆V,X is said to be switching‐equivalent to X⊕∇G(V0), where ∇G(V0) is the set of edges in G each of which has exactly one end in V0 and A⊕B is the symmetric difference of two sets A and B. Lukot'ka and Rollová showed that when G is regular and bipartite, X is nonfeasible if and only if X is switching‐equivalent to ∅. This article extends Lukot'ka and Rollová's result by showing that this conclusion holds as long as G is matching‐covered and bipartite. This article also studies matching‐covered graphs G whose nonfeasible edge sets are switching‐equivalent to ∅ or E and partially characterizes these matching‐covered graphs in terms of their ear decompositions. Another aim of this article is to construct infinite many r‐connected and r‐regular graphs of class 1 containing nonfeasible edge sets not switching‐equivalent to either ∅ or E for an arbitrary integer r with r≥3, which provides a negative answer to a problem proposed by He et al. [ABSTRACT FROM AUTHOR]

Let G=(V,E) be a simple graph with V={1,2,...,n} and χ(G,x) be its chromatic polynomial. For an ordering π=(v1,v2,...,vn) of elements of V, let δG(π) be the number of integers i, where 1≤i≤n−1, with either vi

For a bridgeless graph G, its flow polynomial is defined to be the function F(G,q), which counts the number of nonwhere‐zero Γ‐flows on an orientation of G whenever q is a positive integer and Γ is an additive Abelian group of order q. It was introduced by Tutte in 1950, and the locations of zeros of this polynomial have been studied by many researchers. This paper gives a survey on the results and problems on the study of real zeros of flow polynomials. [ABSTRACT FROM AUTHOR]

For a k $k$‐tree T $T$, a generalization of a tree, the local mean order of sub‐k $k$‐trees of T $T$ is the average order of sub‐k $k$‐trees of T $T$ containing a given k $k$‐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 k $k$‐tree T $T$, does the maximum local mean order of sub‐k $k$‐trees containing a given k $k$‐clique occur at a k $k$‐clique that is not a major k $k$‐clique of T $T$? In this paper, we give it an affirmative answer. [ABSTRACT FROM AUTHOR]

Although accumulating evidence has shown the important function of long non-coding RNAs (lncRNAs) in tumor progression and chemotherapy resistance, the role of lncRNA DLEU1 in regulating proliferation, invasion, and chemoresistance of bladder cancer (BCA) cells remains largely unknown. Here, we found that DLEU1 was upregulated in BLCA tissues and BCA patients with high DLEU1 expression exhibited a shorter survival time. Furthermore, mechanistic analysis and functional assays validated that DLEU1 induced cell proliferation, invasion, and cisplatin resistance of BCA cells by de-repressing the expression of HS3ST3B1 through sponging miR-99b. Low miR-99b and high HS3ST3B1 levels were correlated with worse prognosis in patients with BCA. Ectopic expression of HS3ST3B1 or inhibition of miR-99b reversed DLEU1 knockdown-mediated suppression of cell proliferation, invasion, and cisplatin resistance. Thus, our study revealed a novel role for the DLEU1/miR-99b/HS3ST3B1 axis in regulating proliferation, invasion, and cisplatin resistance of BCA cells. [ABSTRACT FROM AUTHOR]

Background/Aims: FK506 is an immunosuppressive drug and a calcineurin inhibitor that has been widely used in kidney disease in recent years. FK506 shows a wide range of biological and pharmaceutical effects; however, the mechanism of its anti- proliferative effect has not been well elucidated. An IgA nephropathy (IgAN) model was used to generate a mesangial cell proliferation model. This study aims to examine the effect of FK506 on IgAN rats and the underlying mechanisms. Methods: Hematuria, proteinuria and renal function were measured. To observe the pathological conditions, we performed HE (hematoxylin - eosin) and PAS (periodic acid - schiff) staining. Transcription and protein expression levels were detected by qRT - PCR (quantitative real-time polymerase chain reaction) and Wb (western blotting). The location and semi-quantitative expression levels of TRPCs, CaN (Calcineurin) and α-SMA were examined by IHC (Immunohistochemical staining). Results: We found that FK506 could improve hematuria, proteinuria and renal function, especially in the HF (high-dose FK506) groups. Renal pathological changes were ameliorated in the treatment groups. FK506 could significantly decrease TRPCs, CaN, phosphorylation of ERK1/2 and α-SMA expression. Conclusion: Taken together, these results suggest that the therapeutic effect of FK506 on IgAN might be partially associated with the down-regulated expression of TRPC channels, CaN and phosphorylation of ERK1/2. [ABSTRACT FROM AUTHOR]

For any graph G, let [ABSTRACT FROM AUTHOR]

The chromatic equivalence class of a graph G is the set of graphs that have the same chromatic polynomial as G . We find the chromatic equivalence class of the complete tripartite graphs K 1 , n , n + 2 for all n ≥ 2 . This partially answers a question raised in Chia and Ho (2009), which asks for the chromatic equivalence class of the graph K 1 , m , n where 2 ≤ m ≤ n . [ABSTRACT FROM AUTHOR]

A mixed hypergraph H is a triple (X , C , D) , where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ , a proper λ -coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ -colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (− ∞ , 0) and (0 , 1) , which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs. [ABSTRACT FROM AUTHOR]

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps. [ABSTRACT FROM AUTHOR]

Abstract: Let be a connected plane graph, be the corresponding link diagram via medial construction, and be the number of components of the link diagram . In this paper, we first provide an elementary proof that , where is the nullity of . Then we lay emphasis on the extremal graphs, i.e. the graphs with . An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs. [Copyright &y& Elsevier]

A signed edge domination function (or SEDF) of a simple graph G = (V , E) is a function f : E → { 1 , − 1 } such that ∑ e ′ ∈ N [ e ] f (e ′) ≥ 1 holds for each edge e ∈ E , where N [ e ] is the set of edges in G that share at least one endpoint with e. Let γ s ′ (G) denote the minimum value of f (G) among all SEDFs f , where f (G) = ∑ e ∈ E f (e). In 2005, Xu conjectured that γ s ′ (G) ≤ n − 1 , where n is the order of G. This conjecture has been proved for the two cases v o d d (G) = 0 and v e v e n (G) = 0 , where v o d d (G) (resp. v e v e n (G)) is the number of odd (resp. even) vertices in G. This article proves Xu's conjecture for v e v e n (G) ∈ { 1 , 2 }. We also show that for any simple graph G of order n , γ s ′ (G) ≤ n + v o d d (G) ∕ 2 and γ s ′ (G) ≤ n − 2 + v e v e n (G) when v e v e n (G) > 0 , and thus γ s ′ (G) ≤ (4 n − 2) ∕ 3. Our result improves the best current upper bound of γ s ′ (G) ≤ ⌈ 3 n ∕ 2 ⌉. [ABSTRACT FROM AUTHOR]

In this paper, we prove that (− ∞ , 0) is a zero-free interval for chromatic polynomials of a family L 0 of hypergraphs and (0 , 1) is a zero-free interval for chromatic polynomials of a subfamily L 0 ′ of L 0 of hypergraphs. These results extend known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs. [ABSTRACT FROM AUTHOR]

Abstract: The atom–bond connectivity (ABC) index of a graph is defined as where is the edge set and is the degree of vertex of . We give an upper bound for the ABC index of connected graphs with fixed number of vertices, number of edges and maximum degree, and characterize the extremal graphs. From this, we obtain an upper bound and extremal graphs for the ABC index of molecular graphs with fixed number of vertices and number of edges. Then we determine the -vertex unicyclic graphs with the maximum, the second, the third and the fourth maximum ABC indices, and the -vertex bicyclic graphs with the maximum and the second maximum ABC indices respectively for . [Copyright &y& Elsevier]