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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)

Let G = (V , E) be a simple graph with n vertices and m edges, P (G , k) be the chromatic polynomial of G , and P (G , L) be the number of L -colorings of G for any k -assignment L. In this article, we show that when k ≥ m − 1 ≥ 3 , P (G , L) − P (G , k) is bounded below by ((k − m + 1) k n − 3 + (k − m + 3) c 3 k n − 5) ∑ u v ∈ E | L (u) ∖ L (v) | , where c ≥ (m − 1) (m − 3) 8 , and in particular, if G is K 3 -free, then c ≥ ( m − 2 2 ) + 2 m − 3. Consequently, P (G , L) ≥ P (G , k) whenever k ≥ m − 1. [ABSTRACT FROM AUTHOR]

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]

In 2013, Ku and Wong showed that for any partitions μ and μ ′ of a positive integer n with the same first part u and the lexicographic order μ ◃ μ ′ , the eigenvalues ξ μ and ξ μ ′ of the derangement graph Γ n have the property | ξ μ | ≤ | ξ μ ′ | , where the equality holds if and only if u = 3 and all other parts are less than 3. In this article, we obtain an analogous conclusion on the eigenvalues of the perfect matching derangement graph M 2 n of K 2 n by finding a new recurrence formula for the eigenvalues of M 2 n. [ABSTRACT FROM AUTHOR]

For any r -uniform hypergraph H with m (≥2) edges, let P (H , k) and P l (H , k) be the chromatic polynomial and the list-color function of H respectively, and let ρ (H) denote the minimum value of | e ∖ e ′ | among all pairs of distinct edges e , e ′ in H. We will show that if r ≥ 3 , ρ (H) ≥ 2 and m ≥ ρ (H) 3 2 + 1 , then P l (H , k) = P (H , k) holds for all integers k ≥ 2.4 (m − 1) ρ (H) log ⁡ (m − 1). [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]

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]

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 − p K 2 (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. [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]

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is well-known that each 1-planar graph has its minimum degree at most 7. Recently, Biedl (2021) showed that any 1-planar graph with minimum degree 7 has at least 24 vertices. In this paper, we characterize 1-planar graphs with 24 vertices and minimum degree 7. Furthermore, we prove that any 1-planar graph of odd order with minimum degree 7 has at least 29 vertices, and the lower bound is tight. In addition, we also characterize 5-connected 7-regular maximal 1-plane graphs. [ABSTRACT FROM AUTHOR]

Abstract This study planned to explore the effects of M2c macrophages on epithelial-to-mesenchymal transition (EMT) of human renal proximal tubular epithelial cells (HK-2). Human monocytic leukaemia cells were induced by TPA and IL-10 to differentiate M2c macrophages. Subsequently HK-2 cells were co-cultured with the M2c macrophages in Transwell chamber. After 48 h of co-culturing the HK-2 cells were detected in the mRNA and protein expression of E-cadherin, α-SMA and vimentin with RT-PCR, immunofluorescence and Western blot respectively. Besides, the migration ability of the HK-2 cells was estimated with Transwell migration assay. ANOVA was used to compare the difference between groups and Student's t -test to conduct multiple comparisons of two groups. P < 0.05 was considered statistically significant. The results showed that the mRNA and protein expression of α-SMA and vimentin of the HK-2 cells were increased but the E-cadherin decreased significantly after 48 h co-culturing with the M2c macrophages (P < 0.05 or P < 0.01). And the migration ability of HK-2 cells were also increased significantly (P < 0.05). It may be concluded that polarized M2c macrophages may have a promoting effect on the EMT of HK-2 cells. [ABSTRACT FROM AUTHOR]

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 x 0 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 ) − x 0 , 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” f : 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 f = ψ ( M ) , then ∑ x ∈ X f ( 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”. [ABSTRACT FROM AUTHOR]

In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph G = ( V , E ) , the number of totally cyclic orientations of G is equal to the value of | P ( H G , − 1 ) | , where P ( H G , λ ) is the chromatic polynomial of a hypergraph H G which is constructed from G . Finally we show that the multiplicity of root “ 0 ” of P ( H , λ ) may be at least 2 for some connected hypergraphs H , and the multiplicity of root “ 1 ” of P ( H , λ ) may be 1 for some connected and separable hypergraphs H and may be 2 for some connected and non-separable hypergraphs H . [ABSTRACT FROM AUTHOR]

For any graph G, let [ABSTRACT FROM AUTHOR]

A tiered graph G = (V , E) with m tiers is a simple graph with V ⊆ 〚 n 〛 , where 〚 n 〛 = { 1 , 2 , ⋯ , n } , and with a surjective map t from V to 〚 m 〛 such that if v is a vertex adjacent to v ′ in G with v > v ′ , then t (v) > t (v ′). For any ordered partition p = (p 1 , p 2 , ⋯ , p m) of n , let T p denote the set of tiered trees with vertex set 〚 n 〛 and with a map t : 〚 n 〛 → 〚 m 〛 such that | t − 1 (i) | = p i for all i = 1 , 2 , ... , m. For any T ∈ T p , let K T denote the complete tiered graph whose vertex set and tiering map are the same as those of T. If the edges of K T are ordered lexicographically by their endpoints, then the weight w (T) of T is the external activity of T in K T , i.e., the number of edges e ∈ E (K T) ∖ E (T) such that e is the least element in the unique cycle determined by T ∪ e. Let P p (q) = ∑ T ∈ T p q w (T). Dugan, Glennon, Gunnells and Steingrímsson (2019) [4] asked for an elementary proof of the identity P p (q) = P π (p) (q) for any permutation π of 1 , 2 , ⋯ , m , where π (p) = (p π (1) , p π (2) , ⋯ , p π (m)). In this article, we will prove an extension of this identity by applying Tutte polynomials. Furthermore, we also provide a proof of the identity P (1 , p 1 , p 2) (q) = P (p 1 + 1 , p 2 + 1) (q) via Tutte polynomials. [ABSTRACT FROM AUTHOR]

Let G be a complete multi-partite graph of order n. In this paper, we consider the anti-Ramsey number ar (G , T q) with respect to G and the set T q of trees with q edges, where 2 ≤ q ≤ n − 1. For the case q = n − 1 , the result has been obtained by Lu, Meier and Wang. We will extend it to q < n − 1. We first show that ar (G , T q) = ℓ q (G) + 1 , where ℓ q (G) is the maximum size of a disconnected spanning subgraph H of G with the property that any two components of H together have at most q vertices. Using this equality, we obtain the exact values of ar (G , T q) for n − 3 ≤ q ≤ n − 1. Moreover, for the general case when (4 n − 2) / 5 ≤ q ≤ n − 1 , ar (G , T q) can be determined by a simple algorithm. In particular, the explicit expression of ar (G , T q) is given when G has a partite set much larger than all the other partite sets. [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]

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 P D P (G , m) of G , introduced by Kaul and Mudrock in 2019, is a generalization of P (G , m) with P D P (G , m) ≤ P (G , m) for each positive integer m. Let P D P (G) ≈ P (G) (resp. P D P (G) < P (G)) denote the property that P D P (G , m) = P (G , m) (resp. P D P (G , m) < P (G , m)) holds for sufficiently large integers m. It is an interesting problem of finding graphs G for which P D P (G) ≈ P (G) (resp. P D P (G , m) < P (G , m)) holds. Kaul and Mudrock showed that if G has an even girth, then P D P (G) < P (G) and Mudrock and Thomason recently proved that P D P (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 P D P (G) < P (G) holds. However, the converse statement of this conclusion fails with infinitely many counterexamples. We then prove that P D P (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. [ABSTRACT FROM AUTHOR]

• Page 2, the first line below Theorem 1: The set NST u (T) is revised to the set of non-spanning trees T such that u ∈ V (T) and G − V (T) has no isolated vertices. • Add two lemmas in the beginning of Section 3. The purpose is to apply them to simplify the proof of Theorem 3. • The proof of Theorem 3 has been revised according to the suggestion from one of the reviewers to consider the set of spanning digraphs D which have the property that i d D (v n) = 0 and i d D (v i) = 1 for each i ∈ [ n − 1 ]. • In page 8, we add a paragraph for an upper bound of τ (G) and a corollary. It is well-known that the number of spanning trees, denoted by τ (G) , in a connected multi-graph G can be calculated by the Matrix-Tree Theorem and Tutte's deletion-contraction formula. In this short note, we find an alternate method to compute τ (G) by degrees of vertices. [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]

Abstract: Let be a connected graph, where is even. In this paper we show that the line graph of contains at least perfect matchings, and characterize such that has exactly perfect matchings. As applications, we use a unified approach to solve the dimer problem on the Kagomé lattice, lattice, and Sierpinski gasket with dimension two in the context of statistical physics. [Copyright &y& Elsevier]

The article discusses mathematical theorems regarding identities with the help of arbitrary polynomials. It mentions that two theorems have been proposed by mathematician Thomas Dence. It states that identities have been established with the help of Katsuura theorem which was given by scientist Hidefumi Katsuura and highlights that identity in algebra is a set of binary operations that produces identity matrix.

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]

A 1-planar graph is a graph which has a drawing on the plane such that each edge has at most one crossing. Czap and Hudák showed that every 1-planar graph with n vertices has crossing number at most n - 2 . In this paper, we prove that every maximal 1-planar graph G with n vertices has crossing number at most n - 2 - (2 λ 1 + 2 λ 2 + λ 3) / 6 , where λ 1 and λ 2 are respectively the numbers of 2-degree and 4-degree vertices in G, and λ 3 is the number of odd vertices w in G such that either d G (w) ≤ 9 or G - w is 2-connected. Furthermore, we show that every 3-connected maximal 1-planar graph with n vertices and m edges has crossing number m - 3 n + 6 . [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]

A complex unit gain graph (or T -gain graph) is a triple Φ = (G , T , φ) (or (G , φ) for short) consisting of a simple graph G , as the underlying graph of (G , φ) , the set of unit complex numbers T = { z ∈ C : | z | = 1 } and a gain function φ : E → → T with the property that φ (e i , j) = φ (e j , i) − 1. In this paper, we prove that 2 m (G) − 2 c (G) ≤ r (G , φ) ≤ 2 m (G) + c (G) , where r (G , φ) , m (G) and c (G) are the rank of the Hermitian adjacency matrix H (G , φ) , the matching number and the cyclomatic number of G , respectively. Furthermore, the complex unit gain graphs (G , T , φ) with r (G , φ) = 2 m (G) − 2 c (G) and r (G , φ) = 2 m (G) + c (G) are characterized. These results generalize the corresponding known results about undirected graphs, mixed graphs and signed graphs. Moreover, we show that 2 m (G − V 0) ≤ r (G , φ) ≤ 2 m (G) + S holds for any S ⊂ V (G) such that G − S is bipartite and any subset V 0 of V (G) such that G − V 0 is acyclic. [ABSTRACT FROM AUTHOR]

Purpose: This retrospective cohort study aimed to investigate the value of preimplantation genetic testing for aneuploidy (PGT-A) as a screening test for patients suffering from unexplained recurrent implantation failure (RIF). Methods: After screening patients in one reproductive medicine center, twenty-nine, forty-nine and thirty-eight women (< 40 years old) who had suffered unexplained RIF with PGT-A, or RIF without PGT-A, or no RIF with PGT-A were included. The clinical pregnancy rate and live birth rate per transfer, the conservative and optimal cumulative clinical pregnancy rates (CCPR) and live birth rates (CLBR) after three blastocyst FETs were analyzed. Results: The live birth rate per transfer was significantly higher in the RIF + PGT-A group than that in the RIF + NO PGT-A group (47.6% vs. 24.6%, p = 0.014). After 3 cycles of FET, RIF + PGT-A group had significantly higher conservative CLBR and optimal CLBR compared to the RIF + NO PGT-A group (69.0% vs. 32.7%, p = 0.002 and 73.7% vs. 57.5%, p = 0.016), but had similar conservative and optimal CLBRs compared to the NO RIF + PGT-A group. The number of FET cycles required when half women achieved a live birth was 1 in the PGT-A group and 3 in RIF + NO PGT-A group. The miscarriage rates were not different between the RIF + PGT-A and RIF + NO PGT-A, RIF + PGT-A and NO RIF + PGT-A groups. Conclusion: PGT-A did be superior in reducing the number of transfer cycles required to achieve a similar live birth rate. Further studies to identify the RIF patients who would benefit most from PGT-A are necessary. [ABSTRACT FROM AUTHOR]

The E2F transcription factor family, whose members are encoded by the E2F1-E2F8 genes, plays pivotal roles in the cell cycle, apoptosis, metabolism, stemness, metastasis, aging, angiogenesis, tumor promotion or suppression, and other biological processes. The activity of E2Fs is regulated at multiple levels, with posttranslational modifications being an important regulatory mechanism. There are numerous types of posttranslational modifications, among which phosphorylation, acetylation, methylation, ubiquitination, SUMOylation, neddylation, and poly(ADP-ribosyl)ation are the most commonly studied in the context of the E2F family. Posttranslational modifications of E2F family proteins regulate their biological activity, stability, localization, and interactions with other biomolecules, affecting cell proliferation, apoptosis, DNA damage, etc., and thereby playing roles in physiological and pathological processes. Notably, these modifications do not always act alone but rather form an interactive regulatory network. Currently, several drugs targeting posttranslational modifications are being studied or clinically applied, in which the proteolysis-targeting chimera and molecular glue can target E2Fs. This review aims to summarize the roles and regulatory mechanisms of different PTMs of E2F family members in the physiological state and in cancer and to briefly discuss their clinical significance and potential therapeutic use. [Display omitted] • Different posttranslational modifications of E2Fs mediated by various enzymes. • Roles and regulatory mechanisms of posttranslational modifications on E2Fs. • Crosstalk of various posttranslational modifications of E2Fs. • Potential value of E2F posttranslational modifications in cancer therapy. [ABSTRACT FROM AUTHOR]

Increased excitability and contractility of detrusor smooth muscle (DSM) cells are associated with overactive bladder (OAB), which is often induced by obesity. Small-conductance Ca2+-activated K+ (SK) channels regulate the excitability and contractility of DSM cells. Selective pharmacological activation of SK channels attenuates hyperpolarization and the decreased relaxation effect in DSM cells in obesity-induced OAB. However, additional data are needed to confirm the regulatory effect of SK channels on the function of DSM cells in obesity-related OAB. The tested hypothesis was that activation of SK channels decreases modulation of DSM function in a rat model of obesity-related OAB. Female Sprague Dawley rats were fed a normal diet (ND) or a high-fat diet (HFD), weighed after 12 weeks, and subjected to urodynamic study, patch-clamp electrophysiology, and isometric tension recording. The average body weight and incidence of OAB were increased in the HFD group. Patch-clamp studies revealed that pharmacological activation of SK channels with SKA-31 had attenuated hyperpolarization of DSM cells. In addition, isometric tension recordings indicated that SKA-31 decreased relaxation of spontaneous phasic contractions of DSM strips in the HFD group. Attenuated function of SK channels increased the excitability and contractility of DSM cells, which contributed to the occurrence of OAB, suggesting that SK channels are potential therapeutic targets for control of OAB. [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]

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]

A graph G is k -crossing-critical if c r (G) ≥ k , but c r (G ∖ e) < k for each edge e ∈ E (G) , where c r (G) is the crossing number of G. It is known that the latest upper bound of c r (G) for a k -crossing-critical graph G is 2 k + 8 k + 47 when δ (G) ≥ 3 , and 2 k + 35 when δ (G) ≥ 4 , where δ (G) is the minimum degree of G. In this paper, we mainly show that for any k -crossing-critical graph G with n vertices, c r (G) ≤ 2 k + 8 when δ (G) ≥ 4 , and c r (G) ≤ 2 k − k / 2 n + 35 / 6 when δ (G) ≥ 5. [ABSTRACT FROM AUTHOR]