Families of Small Regular Graphs of Girth 5

In this paper we obtain $(q+3)$--regular graphs of girth 5 with fewer vertices than previously known ones for $q=13,17,19$ and for any prime $q \ge 23$ performing operations of reductions and amalgams on the Levi graph $B_q$ of an elliptic semiplane of type ${\cal C}$. We also obtain a 13-regular graph of girth 5 on 236 vertices from $B_{11}$ using the same technique

Locating-dominating codes in cycles

The smallest cardinality of an r-locating-dominating code in a cycle C_n of length n is denoted by M_r^{LD}(C_n). In this paper, we prove that for any r geq 5 and n geq n_r when n_r is large enough (n_r=mathcal{O}(r^3)) we have n/3 leq M_r^{LD}(C_n) leq n/3+1 if n equiv 3 pmod{6} and M_r^{LD}(C_n) = lceil n/3 ceil otherwise. Moreover, we determine the exact values of M_3^{LD}(C_n) and M_4^{LD}(C_n) for all n

On bipartite (1,1,k)(1,1,k)-mixed graphs

Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs, in terms of the number of vertices, are presented for small diameters. Moreover, two infinite families of such graphs with diameter $k$ and number of vertices of the order of $2^{k/2}$ are proposed, one of them being totally regular $(1,1)$-mixed graphs. In addition, we present two more infinite families called chordal ring and chordal double ring mixed graphs, which are bipartite and related to tessellations of the plane. Finally, we give an upper bound that improves the Moore bound for bipartite mixed graphs for $r = z = 1$

Locating-dominating codes in paths

Bertrand, Charon, Hudry and Lobstein studied, in their paper in 2004 [1] r-locating-dominating codes in paths P(n). They conjectured that if r >= 2 is a fixed integer, then the smallest cardinality of an r-locating-dominating code in P(n), denoted by M(r)(LD) (P(n)), satisfies M(r)(LD)(P(n)) = [(n + 1)/3] for infinitely many values of n. We prove that this conjecture holds. In fact, we show a stronger result saying that for any r >= 3 we have M(r)(LD) (P(n)) = [(n + 1)/3] for all n >= n(r), when n(r) is large enough. In addition, we solve a conjecture on location-domination with segments of even length in the infinite path. (C) 2011 Elsevier B.V. All rights reserved

High girth column-weight-two LDPC codes based on distance graphs

Copyright © 2007 G. Malema and M. Liebelt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.LDPC codes of column weight of two are constructed from minimal distance graphs or cages. Distance graphs are used to represent LDPC code matrices such that graph vertices that represent rows and edges are columns. The conversion of a distance graph into matrix form produces an adjacency matrix with column weight of two and girth double that of the graph. The number of 1's in each row (row weight) is equal to the degree of the corresponding vertex. By constructing graphs with different vertex degrees, we can vary the rate of corresponding LDPC code matrices. Cage graphs are used as examples of distance graphs to design codes with different girths and rates. Performance of obtained codes depends on girth and structure of the corresponding distance graphs.Gabofetswe Malema and Michael Liebel