Lower bounds on two-terminal network reliability

Computing the probability that two nodes in a probabilistic network are connected is a well-known computationally difficult problem. Two strategies are devised for obtaining lower bounds on the connection probability for two terminals. The first improves on the Kruskal-Katona bound by using efficient computations of small pathsets. The second strategy employs efficient algorithms for finding edge-disjoint paths. The resulting bounds are compared; while the edge-disjoint path bounds typically outperform the Kruskal-Katona bounds, they do not always do so. Finally, a method is outlined for developing a uniform bound which combines both strategies.     

Lower bounds and upper bounds for chromatic polynomials

In this paper we give lower bounds and upper bounds for chromatic polynomials of simple undirected graphs on n vertices having m edges and girth exceeding g © 1993 John Wiley & Sons, Inc.     

Lower bounds for transition probabilities on graphs

The paper presents two results. The first one provides separate conditions for the upper and lower estimates of the distribution of the time of exit from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.     

Lower bounds for combinatorial problems on graphs

Nontrivial lower bounds are given for the computation time of various combinatorial problems on graphs under a linear or algebraic decision tree model. An $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is obtained for a “travelling salesman problem” on a weighted complete graph of n vertices. Moreover it is shown that the same bound is valid for the “subgraph detection problem” with respect to property P if P is hereditary and determined by components. Thus an $\text{Ω(n}{}^3\text{log}\text{n)}$ bound is established in a unified way for a rather large class of problems.     

Lower bounds on fluctuations for internal DLA

We consider internal diffusion limited aggregation in dimension two or more. This is a random cluster growth model, where random walks start at the origin of the lattice, one at a time, and stop moving when reaching a site that is not occupied by previous walks. When $n$ random walks are sent from the origin, we establish a lower bound for the inner and outer errors fluctuations of order square root of the logarithm of $n$ . When dimension is three or more, this lower bound matches the upper bound recently obtained in independent works of Asselah and Gaudillière (Ann Prob arXiv:1011.4592, 2010) and Jerison et al. (Internal DLA and the Gaussian free field arXiv:1012.3453, 2010). Also, we produce as a corollary of our proof of Asselah and Gaudillière (Ann Prob arXiv:1011.4592, 2010), an upper bound for the fluctuation of the inner error in a specified direction.     

Lower bounds for Seshadri constants

One of Demailly's characterization of Seshadri constants on ample line bundles works with Lelong numbers of certain positive singular hermitian metrics. In this note this is translated into algebraic terms by using sections of multiples of the line bundle. The resulting formula for Seshadri constants is applied to compute Seshadri constants on blown up products of curves, to disprove a conjectured characterization of maximal rationally connected quotients and to introduce a new approach to Nagata's conjecture. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lower bounds for critical values

We prove here a Filippov type implicit function theorem for Carath$eacute;odory mappings in general topological space. We also prove a Banach space version which is applicable to optimal control problems. For this purpose, we introduce the basic concepts of separation in topological spaces. Connections between separation properties of spaces and measurability of graphs of multifunctions are established.

AMS (MOS): 57C65     

Lower bounds on the cover-index of a graph

In this paper we give some new lower bounds for the cover-index of graphs with multiple edges permitted. The results are analogous to upper bounds for the chromatic index. We show that a simple graph with cover-index different from the minimum degree has at least three vertices of minimum degree. This implies that almost all simple graphs have cover-index equal to the minimum degree.     

Lower bounds on covering codes via partition matrices

Let Kq(n,R) denote the minimal cardinality of a q-ary code of length n and covering radius R. Let σq(n,s;r) denote the minimal cardinality of a q-ary code of length n, which is s-surjective with radius r. In order to lower-bound Kq(n,n−2) and σq(n,s;s−2) we introduce partition matrices and their transversals. Our approach leads to a short new proof of a classical bound of Rodemich on Kq(n,n−2) and to the new bound Kq(n,n−2)?3q−2n+2, improving the first iff 5?n<q?2n−4. We determine Kq(q,q−2)=q−2+σ₂(q,2;0) if q?10. Moreover, we obtain the new powerful recursive bound Kq+1(n+1,R+1)?min{2(q+1),Kq(n,R)+1}.