An algorithm for calculating the independence and vertex-cover polynomials of a graph

The independence polynomial, ω(G,x)=∑w_kx^k, of a graph, G, has coefficients, w_k, that enumerate the ways of selecting k vertices from G so that no two selected vertices share an edge. The independence number of G is the largest value of k for which w_k≠0. Little is known of less straightforward relationships between graph structure and the properties of ω(G,x), in part because of the difficulty of calculating values of w_k for specific graphs. This study presents a new algorithm for these calculations which is both faster than existing ones and easily adaptable to high-level computer languages.     

An Ore-type analogue of the Sauer-Spencer Theorem

Two graphs G₁ and G₂ of order n pack if there exist injective mappings of their vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer proved that if Δ (G₁) Δ (G₂) < 0.5n, then G₁ and G₂ pack. In this note, we study an Ore-type analogue of the Sauer–Spencer Theorem. Let θ(G) = max{d(u) + d(v): uv∈E(G)}. We show that if θ(G₁)Δ(G₂) < n, then G₁ and G₂ pack. We also characterize the pairs (G₁,G₂) of n-vertex graphs satisfying θ(G₁)Δ(G₂) = n that do not pack. This work was supported in part by NSF grant DMS-0400498. The work of the first author was also partly supported by grant 05-01-00816 of the Russian Foundation for Basic Research.     

Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

We introduce a generalized dot product and provide some embedding conditions under which the genus of a graph does not rise under a dot product with the Petersen graph. Using these conditions, we disprove a conjecture of Tinsley and Watkins on the genus of dot products of the Petersen graph and show that both Grünbaum’s Conjecture and the Berge-Fulkerson Conjecture hold for certain infinite families of snarks. Additionally, we determine the orientable genus of four known snarks and two known snark families, construct a new example of an infinite family of snarks on the torus, and construct ten new examples of infinite families of snarks on the 2-holed torus; these last constructions allow us to show that there are genus-2 snarks of every even order n ≥ 18.     

Splittings of generalized Baumslag–Solitar groups

We study the structure of generalized Baumslag–Solitar groups from the point of view of their (usually non-unique) splittings as fundamental groups of graphs of infinite cyclic groups. We find and characterize certain decompositions of smallest complexity (fully reduced decompositions) and give a simplified proof of the existence of deformations. We also prove a finiteness theorem and solve the isomorphism problem for generalized Baumslag–Solitar groups with no non-trivial integral moduli.      

连通的顶点可迁图的色唯一性

本文给出从一个已知的顶点可迁的非色唯一图出发,构造无穷多个顶点可迁的非色唯一图的一种方法,据此给出若干类无穷多个连通的顶点可迁,但不是色唯一的图簇,从而进一步否定地回答了Chia在[1]中提出的问题.     

A Sharp Upper Bound for the Number of Spanning Trees of a Graph

Let G = (V,E) be a simple graph with n vertices, e edges and d₁ be the highest degree. Further let λ_i, i = 1,2,...,n be the non-increasing eigenvalues of the Laplacian matrix of the graph G. In this paper, we obtain the following result: For connected graph G, λ₂ = λ₃ = ... =  λ_n-1 if and only if G is a complete graph or a star graph or a (d₁,d₁) complete bipartite graph. Also we establish the following upper bound for the number of spanning trees of G on n, e and d₁ only:

Several improved asymptotic normality criteria and their applications to graph polynomials

As widely regarded, one of the most classical and remarkable tools to measure the asymptotic normality of combinatorial statistics is due to Harper's real-rooted method proposed in 1967. However, this classical theorem exists some obvious shortcomings, for example, it requests all the roots of the corresponding generating function, which is impossible in general.Aiming to overcome this shortcoming in some extent, in this paper we present an improved asymptotic normality criterion, along with several variant versions, which usually just ask for one coefficient of the generating function, without knowing any roots. In virtue of these new criteria, the asymptotic normality of some usual combinatorial statistics can be revealed and extended. Among which, we introduce the applications to matching numbers and Laplacian coefficients in detail. Some relevant conjectures, proposed by Godsil (Combinatorica, 1981) and Wang et al. (J. Math. Anal. Appl., 2017), are generalized and verified as corollaries.     

Graph theoretical duality perspective on conjugate structures and its applications

The paper shows that representing structures, beams and frames by mathematical models based on graph theory enables to provide new perspective on known conjugate structure theorems in mechanics. It is shown that the latter theorems can be derived from the graph theoretical duality principle applied upon the graph representations of the structures. The results reported indicate upon theoretical value of the approach, as the established mathematical foundation can be employed in a variety of mechanical disciplines.     

CAMWI: Detecting protein complexes using weighted clustering coefficient and weighted density

Detection of protein complexes is very important to understand the principles of cellular organization and function. Recently, large protein–protein interactions (PPIs) networks have become available using high-throughput experimental techniques. These networks make it possible to develop computational methods for protein complex detection. Most of the current methods rely on the assumption that protein complex as a module has dense structure. However complexes have core-attachment structure and proteins in a complex core share a high degree of functional similarity, so it expects that a core has high weighted density. In this paper we present a Core-Attachment based method for protein complex detection from Weighted PPI Interactions using clustering coefficient and weighted density. Experimental results show that the proposed method, CAMWI improves the accuracy of protein complex detection.