13 research outputs found
Computational determination of (3,11) and (4,7) cages
A (k,g)-graph is a k-regular graph of girth g, and a (k,g)-cage is a
(k,g)-graph of minimum order. We show that a (3,11)-graph of order 112 found by
Balaban in 1973 is minimal and unique. We also show that the order of a
(4,7)-cage is 67 and find one example. Finally, we improve the lower bounds on
the orders of (3,13)-cages and (3,14)-cages to 202 and 260, respectively. The
methods used were a combination of heuristic hill-climbing and an innovative
backtrack search
Families of Small Regular Graphs of Girth 5
In this paper we obtain --regular graphs of girth 5 with fewer
vertices than previously known ones for and for any prime performing operations of reductions and amalgams on the Levi graph of
an elliptic semiplane of type . We also obtain a 13-regular graph of
girth 5 on 236 vertices from using the same technique
Finding small regular graphs of girths 6, 8 and 12 as subgraphs of cages
AbstractSmall k-regular graphs of girth g where g=6,8,12 are obtained as subgraphs of minimal cages. More precisely, we obtain (k,6)-graphs on 2(kq−1) vertices, (k,8)-graphs on 2k(q2−1) vertices and (k,12)-graphs on 2kq2(q2−1), where q is a prime power and k is a positive integer such that q≥k≥3. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth g=6,8,12
An explicit formula for obtaining -cages and others small regular graphs of girth 8
Let be a prime power; -cages have been constructed as incidence
graphs of a non-degenerate quadric surface in projective 4-space . The
first contribution of this paper is a construction of these graphs in an
alternative way by means of an explicit formula using graphical terminology.
Furthermore by removing some specific perfect dominating sets from a
-cage we derive -regular graphs of girth 8 for and ,
having the smallest number of vertices known so far
From Cages to Trapping Sets and Codewords: A Technique to Derive Tight Upper Bounds on the Minimum Size of Trapping Sets and Minimum Distance of LDPC Codes
Cages, defined as regular graphs with minimum number of nodes for a given
girth, are well-studied in graph theory. Trapping sets are graphical structures
responsible for error floor of low-density parity-check (LDPC) codes, and are
well investigated in coding theory. In this paper, we make connections between
cages and trapping sets. In particular, starting from a cage (or a modified
cage), we construct a trapping set in multiple steps. Based on the connection
between cages and trapping sets, we then use the available results in graph
theory on cages and derive tight upper bounds on the size of the smallest
trapping sets for variable-regular LDPC codes with a given variable degree and
girth. The derived upper bounds in many cases meet the best known lower bounds
and thus provide the actual size of the smallest trapping sets. Considering
that non-zero codewords are a special case of trapping sets, we also derive
tight upper bounds on the minimum weight of such codewords, i.e., the minimum
distance, of variable-regular LDPC codes as a function of variable degree and
girth
Incidence matrices of projective planes and of some regular bipartite graphs of girth 6 with few vertices
Let q be a prime power and r=0,1...,q−3. Using the Latin squares obtained by
multiplying each entry of the addition table of the Galois field of order q by an element
distinct from zero, we obtain the incidence matrices of projective planes and the incidence matrices of (q−r)-regular bipartite graphs of girth 6 and −rq−1 vertices in each partite set. Moreover, in this work two Latin squares of order q−1 with entries belonging to {0,1,..., q}, not necessarily the same, are defined to be quasi row-disjoint if and only if the cartesian product of any two rows contains at most one pair (χ,χ) with χ≠0. Using
these quasi row-disjoint Latin squares we find (q−1)-regular bipartite graphs of girth 6 with −q−2 vertices in each partite set. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth 6.Postprint (published version
A construction of small regular bipartite graphs of girth 8
Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present
a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.Postprint (published version
Small regular graphs of girth 7
In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain (q + 1)-regular graphs of girth 7 and order 2q(3) + q(2) + 2q for each even prime power q >= 4, and of order 2q(3) + 2q(2) q + 1 for each odd prime power q >= 5.Postprint (published version