Recently, Zheng and Wu defined the concept of odd spanning tree of a graph, meaning a spanning tree in which every vertex has odd degree. Similar to Cayley’s formula, Feng, Chen and Wu counted the number of odd spanning trees in complete graphs via Prüfer code and the exponential generating function. In this note, we prove the bipartite analogue of a classical spanning tree enumeration formula. By using them and the Boolean function, we give simple proofs for the number of odd spanning trees of complete graphs and complete bipartite graphs.

Counting spanning trees in graphs and networks is an attractive topic to mathematicians and statistical physicists. Very recently, Li and Yan considered the problem of counting spanning trees with one perfect matching in linear hexagonal chains. Lai and Zhu further considered linear (4k+2) -chains. We extend these results to helicene (4k+1) -polygonal chains which commonly appear in molecular and crystal structures.

Let G be a connected graph, and denote by a(G) the number of perfect matchings in G. Let L(G) (M(G)) be the line graph (middle graph) of G respectively. For a graph G ' constructed by adding a pendant edge to each vertex in G, Lai et al. [Discrete Mathematics, 347 (2024) 113847] proved that when the number of vertices in M(G) is even and the maximum degree of G is less than or equal to 4, then a(L(G ')) = a(M(G)) = 2m-n+13 m-n 2,where m and n are the number of edges and vertices in G. The present work generalizes the above result by introducing a broader family of graphs {Gr}. Suppose G is a graph with vertex set V(G) = {u1, ... , un} and let r= (r1, ... , rn) be a vector of positive integers. The graph Gr is obtained from G by adding ri pendant vertices adjacent to each vertex ui in G, and we denote r =& sum;ni=1 ri. We prove that: If |E(Gr)| = m + r is even, then a(L(Gr)) >= 2m-n +13 m+r-2n 2 . When the maximum degree of Gris less than or equal to 5, the equality holds. As applications, we provide explicit counting results for perfect matchings in chain and cyclic silicate structures.

We present a determinantal formula for the number of spanning trees of a complete multipartite graph containing a given spanning forest. Our approach relies on the Generalized Matrix Determinant Lemma and Jacobi's formula for the derivative of a determinant. This work generalizes known results for complete bipartite graphs and offers an algebraic perspective on the problem. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Let Kn be the complete graph of order n. Very recently, the number of spanning trees (the NST for short) and the resistance distances in Kn-chain (ring) graphs were determined explicitly. We generalized the concept to the generalized Kn-chain (ring) graph [Lns]m ([Cns]m). New formulae for the NST of [Lns]m and [Cns]m were given by a simple and more physical way with a novel technique of adding a pair of positive and negative edges, avoiding complicated linear algebraic computations.

In 1964, Cayley's formula was generalized to a fascinating formula on the number of spanning trees of Kn containing a given spanning forest by J.W. Moon. It was not until 58 years later that the second Moon-type formula was discovered for Km,n by Dong and the author. In this note, we obtain weighted versions for these two Moon-type formulae with much shorter proofs. (c) 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Visibility representation of digraphs was introduced by Axenovich et al. (2013) as a natural generalization of t-bar visibility representation of undirected graphs. A t-bar visibility representation of a digraph G assigns each vertex at most t horizontal bars in the plane so that there is an arc xy in the digraph if and only if some bar for x "sees" some bar for y above it along an unblocked vertical strip with positive width. The visibility number b(G) is the least t such that G has a t-bar visibility representation. In this paper, we solve several problems about b(G) posed by Axenovich et al. and prove that determining whether the bar visibility number of a digraph is 2 is NP-complete. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

For a subgraph G of a complete graph K_n , the K_n -complement of G, denoted by K_n-G , is the graph obtained from K_n-G by removing all the edges of G. In this paper, we express the number of spanning trees of the K_n -complement K_n-G of a bipartite graph G in terms of the determinant of the biadjcency matrices of all induced balanced bipartite subgraphs of G, which are nonsingular, and we derive formulas of the number of spanning trees of K_n-G for various important classes of bipartite graphs G, some of which generalize some previous results.

In 1964, Moon extended Cayley's formula to a nice expression of the number of spanning trees in complete graphs containing any fixed spanning forest. After nearly 60 years, Dong and the first author discovered the second Moon-type formula: an explicit formula of the number of spanning trees in complete bipartite graphs containing any fixed spanning forest. Followed this direction, Li, Chen and Yan found the Moon-type formula for complete 3- and 4-partite graphs. These are the only families of graphs that have the corresponding Moon-type formulas. In this paper, we first determine resistance distances in the vertex-weighted complete split graph S omega m , n . Then we obtain the Moon-type formula for the vertex-weighted complete split graph S omega m , n , that is, the weighted spanning tree enumerator of S omega m , n containing any fixed spanning forest. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

The problem of counting spanning trees of graphs or networks is a fundamental and crucial area of research in combinatorics, while has numerous important applications in statistical physics, network theory and theoretical computer science. Very recently, Kosar, Zaman, Ali and Ullah obtained a nice formula on the number of spanning trees of a K-5-chain network K-5(& ell;) constructed by connecting & ell;copys of complete graphs K-5. They made extensive use of matrix theory and spectral graph theory, especially the normalized Laplacian of graphs. In this paper, by using a rather simple and more physical treatment (the mesh-star transformation in electrical network) without any linear algebra, we generalize their result to K-n-chain graphs and K-n-ring graphs. The results show that there is a simple relation between the number of spanning trees of the K-n-chain graph L-n(& ell;) and the K-n-ring graph C-n(& ell;). We also calculate the corresponding tree entropy (or so called 'the asymptotic growth constant') and find that the tree entropy of the corresponding K-n-chain graphs and K-n-ring graphs are totally the same.

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 K m , 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 K m , n ${K}_{m,n}$ with components T 1 , T 2 , … , T k ${T}_{1},{T}_{2},\ldots ,{T}_{k}$ , the number of spanning trees in K m , n ${K}_{m,n}$ which contain all edges in F $F$ is equal to 1 m n ∏ i = 1 k ( m i n + n i m ) 1 − ∑ i = 1 k m i n i m i n + n i m , $\frac{1}{mn}\left(\prod _{i=1}^{k}({m}_{i}n+{n}_{i}m)\right)\left(1-\sum _{i=1}^{k}\frac{{m}_{i}{n}_{i}}{{m}_{i}n+{n}_{i}m}\right),$ where m i = ∣ V ( T i ) ∩ X ∣ ${m}_{i}=| V({T}_{i})\cap X| $ and n i = ∣ V ( T i ) ∩ Y ∣ ${n}_{i}=| V({T}_{i})\cap Y| $ for i = 1 , 2 , … , k $i=1,2,\ldots ,k$ .

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.

Connectivity is a critical parameter which can measure the reliability of networks. Let [Formula: see text] be a vertex set of [Formula: see text]. If [Formula: see text] has at least [Formula: see text] components, then [Formula: see text] is a [Formula: see text]-component cut of [Formula: see text]. The [Formula: see text]-component connectivity [Formula: see text] of [Formula: see text] is the vertex number of a smallest [Formula: see text]-component cut. Cartesian product of graphs is a useful method to construct a large network. We will use Cauchy–Schwarz inequality to determine the component connectivity of Cartesian product of some graphs.

In this short note, we construct an infinite family of counterexamples to a conjecture on the lower bound of the signed edge domination number of 2-connected graphs. We propose two problems in order to revise the original conjecture.

In 2019, Ye and Yan computed the effective resistances in the nearly balanced complete bipartite graph K n , n − p K 2 ( p ≤ n ). Then the result was extended to K m , n − p K 2 ( p ≤ min { m , n } ) very recently. In this paper, we obtain the effective resistances and the number of spanning trees in any complete bipartite graph plus a matching.

The chromatic polynomialP(G,x)of a graphGof orderncan be expressed as n-ary sumation i=1n(-1)n-iaixi, whereaiis interpreted as the number of broken-cycle-free spanning subgraphs ofGwith exactlyicomponents. The parameter epsilon(G)= n-ary sumation i=1n(n-i)ai/ n-ary sumation i=1naiis the mean size of a broken-cycle-free spanning subgraph ofG. In this article, we confirm and strengthen a conjecture proposed by Lundow and Markstrom in 2006 that epsilon(Tn)<epsilon(G)<epsilon(Kn)holds for any connected graphGof ordernwhich is neither the complete graphKnnor a treeTnof ordern. The most crucial step of our proof is to obtain the interpretation of allai's by the number of acyclic orientations ofG.

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 γs′(G) denote the minimum value of f(G) among all SEDFs f, where f(G)=∑e∈Ef(e). In 2005, Xu conjectured that γs′(G)≤n−1, where n is the order of G. This conjecture has been proved for the two cases vodd(G)=0 and veven(G)=0, where vodd(G) (resp. veven(G)) is the number of odd (resp. even) vertices in G. This article proves Xu's conjecture for veven(G)∈{1,2}. We also show that for any simple graph G of order n, γs′(G)≤n+vodd(G)∕2 and γs′(G)≤n−2+veven(G) when veven(G)>0, and thus γs′(G)≤(4n−2)∕3. Our result improves the best current upper bound of γs′(G)≤⌈3n∕2⌉.

In this paper, we first present spectral conditions for the existence of Cn-1 in graphs (2-connected graphs) of order n, which are motivated by a conjecture of Erdos. We also prove spectral conditions for the existence of Hamilton cycles in balanced bipartite graphs. This result presents a spectral analog of Moon-Moser's theorem on Hamilton cycles in balanced bipartite graphs, and extends a previous theorem due to Li and the second author for n sufficiently large. We conclude this paper with two problems on tight spectral conditions for the existence of long cycles of given lengths.

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)=Km,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.

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. It was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total pi-electron energy. But this topological index had not received further attention until 2015. Very recently, Furtula and Gutman [B. Furula, I. Gutman, A forgotten topological index, J. Math. Chem. 53(4) (2015) 1184-1190] reinvestigated the index and named it the "forgotten topological index" or "F-index". There they presented some basic properties of this index and showed that this index can significantly enhance the physico-chemical applicability of the first Zagreb index. In this paper, we study the F-index of bipartite graphs of order n with diameter d and obtain sharp upper bounds for it. As a consequence, bipartite graphs with the largest, second-largest and smallest F-index are characterized.