L(d,2,1)-Labeling of Simple Graphs
An L(3, 2, 1)-labeling is a simplified model for the channel assignment problem. It is a natural generalization of the widely studied L(2, 1)labeling. An L(3, 2, 1)-labeling of a graph G is a function f from the vertex set V (G) to the set of positive integers such that for any two vertices x, y, if d(x, y) = 1, then |f(x) − f(y)| ≥ 3; if d(x, y) = 2, then |f(x) − f(y)| ≥ 2; and if d(x, y) = 3, then |f(x) − f(y)| ≥ 1. The L(3, 2, 1)-labeling number k(G) of G is the smallest positive integer k such that G has an L(3, 2, 1)-labeling with k as the maximum label. In this paper we determine the L(3, 2, 1)-labeling number for paths, cycles, caterpillars, nary trees, complete graphs and complete bipartite graphs. We also present an upper bound for k(G) in terms of the maximum degree of G.
L(d,2,1)-Labeling of Star and Sun Graphs
For positive integer d, L( d ,2,1)-labeling of a graph G is a function f from V(G) to the positive integers, f: V(G )-> {1, 2,…} such that | f(u) – f(v) | ³ d if the distance between any 2 vertices u and v is 1 (D( u,v ) =1), | f(u) – f(v )| ³ 2 if  D( u,v ) = 2, and | f(u) – f(v) | ³ 1 if D( u,v ) = 3. The L( d, 2,1)-labeling number k d (G) of a graph G is the smallest positive integer k d such that G has an L( d ,2,1)-labeling with k d as the maximum label. This paper presents a general k d -value of stars K 1,n and k d -value of sun graphs S n for d = 3. Keywords: L( d ,2,1)-Labeling, distance, L( d ,2,1)-Labeling number, sun graphs, stars.
An automated retinal image quality grading algorithm
This paper introduces an algorithm for the automated assessment of retinal fundus image quality grade. Retinal image quality grading assesses whether the quality of the image is sufficient to allow diagnostic procedures to be applied. Automated quality analysis is an important preprocessing step in algorithmic diagnosis, as it is necessary to ensure that images are sufficiently clear to allow pathologies to be visible. The algorithm is based on standard recommendations for quality analysis by human screeners, examining the clarity of retinal vessels within the macula region. An evaluation against a reference standard data-set is given; it is shown that the algorithm's performance correlates closely with that of clinicians manually grading image quality.