Unbalanced Bipartite Graphs Research Articles

Relations between ordinary energy and energy of a self-loop graph

Let G be a graph on n vertices with vertex set V(G) and let S⊆V(G) with |S|=α. Denote by GS, the graph obtained from G by adding a self-loop at each of the vertices in S. In this note, we first give an upper bound and a lower bound for the energy of GS (E(GS)) in terms of ordinary energy (E(G)), order (n) and number of self-loops (α). Recently, it is proved that for a bipartite graph GS, E(GS)≥E(G). Here we show that this inequality is strict for an unbalanced bipartite graph GS with 0<α<n. In other words, we show that there exits no unbalanced bipartite graph GS with 0<α<n and E(GS)=E(G).

Fast and Perfect Sampling of Subgraphs and Polymer Systems

We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets ) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.

Solving the parametric bipartite maximum flow problem in unbalanced and closure bipartite graphs

Given a directed bipartite graph $$G=(V,E)$$ with a node set $$V=\{s\} \cup V_1 \cup V_2 \cup \{t\}$$ , and an arc set $$E=E_1\cup E_2\cup E_3$$ , where $$E_1=\{s\} \times V_1$$ , $$E_2= V_1 \times V_2$$ , and $$E_3=V_2 \times \{t\}$$ . Chen (1995) presented an $$O(|V| |E| \log (\frac{|V|^2}{|E|}))$$ time algorithm to solve the parametric bipartite maximum flow problem. In this paper, we assume all arcs in $$E_2$$ have infinite capacity [such a graph is called closure graph Hochbaum (1998)], and present a new approach to solve the problem, which runs in $$O(|V_1| |E| \log (\frac{|V_1|^2}{|E|}+2))$$ time using Gallo et al’s parametric maximum flow algorithm, see Gallo et al. (1989). In unbalanced bipartite graphs, we have $$|V_1| << |V_2|$$ , so, our algorithm improves Chens’s algorithm in unbalanced and closure bipartite graphs.

Semisupervised synthetic aperture radar image segmentation with multilayer superpixels

Image segmentation plays a significant role in synthetic aperture radar (SAR) image processing. However, SAR image segmentation is challenging due to speckle. We propose a semisupervised bipartite graph method for segmentation of an SAR image. First, the multilayer over-segmentation of the SAR image, referred to as superpixels, is computed using existing segmentation algorithms. Second, an unbalanced bipartite graph is constructed in which the correlation between pixels is replaced by the texture similarity between superpixels, to reduce the dimension of the edge matrix. To also improve efficiency, we define a new method, called the combination of the Manhattan distance and symmetric Kullback–Leibler divergence, to measure texture similarity. Third, by the Moore–Penrose inverse matrix and semisupervised learning, we construct an across-affinity matrix. A quantitative evaluation using SAR images shows that the new algorithm produces significantly high-quality segmentations as compared with state-of-the-art segmentation algorithms.

Long cycles in unbalanced bipartite graphs

Let G[X,Y] be a 2-connected bipartite graph with |X|≥|Y|. For S⊆V(G), we define δ(S;G):=min{dG(v):v∈S}. We define σ1,1(G):=min{dG(x)+dG(y):x∈X,y∈Y,xy∉E(G)} and σ2(X):=min{dG(x)+dG(y):x,y∈X,x≠y}. We denote by c(G) the length of a longest cycle in G. Jackson [B. Jackson, Long cycles in bipartite graphs, J. Combin. Theory Ser. B 38 (1985) 118–131] proved that c(G)≥min{2δ(X;G)+2δ(Y;G)−2,4δ(X;G)−4,2|Y|}. In this paper, we extend this result, and prove that c(G)≥min{2σ1,1(G)−2,2σ2(X)−4,2|Y|}.

Degree Sum Conditions for Cyclability in Bipartite Graphs

We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d G (v) be the degree of a vertex v in a graph G. For G[X, Y] and $${S \subseteq V(G),}$$ we define $${\sigma_{1,1}(S):=\min\{d_G(x)+d_G(y) : (x,y) \in (X \cap S,Y) \cup (X, Y \cap S), xy \not\in E(G)\}}$$ . Amar et al. (Opusc. Math. 29:345---364, 2009) obtained ? 1,1(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| ? |Y| and $${S \subseteq V(G)}$$ such that ? 1,1(S) ? |X| + 1, then either there exists a cycle containing S or $${|S \cap X| > |Y|}$$ and there exists a cycle containing Y. This degree sum condition is sharp.

Almost Euclidean subspaces of ℓ 1 N VIA expander codes

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X ⊆ ℝN of dimension (1 - o(1))N such that for every x ∈ X, {display equation}. If we are allowed to use N1/log log N ≤ N°(1) random bits and dim(X) ≥ (1 - η)N for any fixed constant η, the lower bound can be further improved to (log N)-°(1) √N||x||2. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

Lossless Condensers, Unbalanced Expanders, And Extractors

Trevisan showed that many pseudorandom generator constructions give rise to constructions of explicit extractors. We show how to use such constructions to obtain explicit lossless condensers. A lossless condenser is a probabilistic map using only O(logn) additional random bits that maps n bits strings to poly(logK) bit strings, such that any source with support size K is mapped almost injectively to the smaller domain. Our construction remains the best lossless condenser to date. By composing our condenser with previous extractors, we obtain new, improved extractors. For small enough min-entropies our extractors can output all of the randomness with only O(logn) bits. We also obtain a new disperser that works for every entropy loss, uses an O(logn) bit seed, and has only O(logn) entropy loss. This is the best disperser construction to date, and yields other applications. Finally, our lossless condenser can be viewed as an unbalanced bipartite graph with strong expansion properties.

Triangle-Free Hypergraphs

In this paper, we give sharp upper bounds on the maximum number of edges in very unbalanced bipartite graphs not containing any cycle of length 6. To prove this, we estimate roughly the sum of the sizes of the hyperedges in triangle-free multi-hypergraphs.

Intersection Representation of Complete Unbalanced Bipartite Graphs

Ap-intersectionrepresentation of a graphGis a map,f, that assigns each vertex a subset of {1,2,…,t} such that {u,v} is an edge if and only if |f(u)∩f(v)|⩾p. The symbolθp(G) denotes this minimumtsuch that ap-intersection representation ofGexists. In 1966 Erdős, Goodman, and Pósa showed that for all graphsGon 2nvertices,θ1(G)⩽θ1(Kn,n)=n2. In 1992, Chung and West conjectured that for all graphsGon 2nvertices,θp(G)⩽θp(Kn,n) whenp⩾1. Subsequently, upper and lower bounds forθp(Kn,n) have been found to be (n2/p)(1+o(1)). We show in this paper that many complete unbalanced bipartite graphs on 2nvertices have a largerp-intersection number thanKn,n. For example, whenp=2,θ2(Kn,n)⩽12n2(1+o(1))<4172n2(1+o(1))⩽θ2(K(5/6)n,(7/6)n).