370,694 research outputs found
On graphs of defect at most 2
In this paper we consider the degree/diameter problem, namely, given natural
numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of
vertices in a graph of maximum degree {\Delta} and diameter D. In this context,
the Moore bound M({\Delta},D) represents an upper bound for N({\Delta},D).
Graphs of maximum degree {\Delta}, diameter D and order M({\Delta},D), called
Moore graphs, turned out to be very rare. Therefore, it is very interesting to
investigate graphs of maximum degree {\Delta} \geq 2, diameter D \geq 1 and
order M({\Delta},D) - {\epsilon} with small {\epsilon} > 0, that is,
({\Delta},D,-{\epsilon})-graphs. The parameter {\epsilon} is called the defect.
Graphs of defect 1 exist only for {\Delta} = 2. When {\epsilon} > 1,
({\Delta},D,-{\epsilon})-graphs represent a wide unexplored area. This paper
focuses on graphs of defect 2. Building on the approaches developed in [11] we
obtain several new important results on this family of graphs. First, we prove
that the girth of a ({\Delta},D,-2)-graph with {\Delta} \geq 4 and D \geq 4 is
2D. Second, and most important, we prove the non-existence of
({\Delta},D,-2)-graphs with even {\Delta} \geq 4 and D \geq 4; this outcome,
together with a proof on the non-existence of (4, 3,-2)-graphs (also provided
in the paper), allows us to complete the catalogue of (4,D,-{\epsilon})-graphs
with D \geq 2 and 0 \leq {\epsilon} \leq 2. Such a catalogue is only the second
census of ({\Delta},D,-2)-graphs known at present, the first being the one of
(3,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2 [14]. Other
results of this paper include necessary conditions for the existence of
({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 4, and the
non-existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 5
such that {\Delta} \equiv 0, 2 (mod D).Comment: 22 pages, 11 Postscript figure
On bipartite graphs of defect at most 4
We consider the bipartite version of the degree/diameter problem, namely,
given natural numbers {\Delta} \geq 2 and D \geq 2, find the maximum number
Nb({\Delta},D) of vertices in a bipartite graph of maximum degree {\Delta} and
diameter D. In this context, the Moore bipartite bound Mb({\Delta},D)
represents an upper bound for Nb({\Delta},D). Bipartite graphs of maximum
degree {\Delta}, diameter D and order Mb({\Delta},D), called Moore bipartite
graphs, have turned out to be very rare. Therefore, it is very interesting to
investigate bipartite graphs of maximum degree {\Delta} \geq 2, diameter D \geq
2 and order Mb({\Delta},D) - \epsilon with small \epsilon > 0, that is,
bipartite ({\Delta},D,-\epsilon)-graphs. The parameter \epsilon is called the
defect. This paper considers bipartite graphs of defect at most 4, and presents
all the known such graphs. Bipartite graphs of defect 2 have been studied in
the past; if {\Delta} \geq 3 and D \geq 3, they may only exist for D = 3.
However, when \epsilon > 2 bipartite ({\Delta},D,-\epsilon)-graphs represent a
wide unexplored area. The main results of the paper include several necessary
conditions for the existence of bipartite -graphs; the complete
catalogue of bipartite (3,D,-\epsilon)-graphs with D \geq 2 and 0 \leq \epsilon
\leq 4; the complete catalogue of bipartite ({\Delta},D,-\epsilon)-graphs with
{\Delta} \geq 2, 5 \leq D \leq 187 (D /= 6) and 0 \leq \epsilon \leq 4; and a
non-existence proof of all bipartite ({\Delta},D,-4)-graphs with {\Delta} \geq
3 and odd D \geq 7. Finally, we conjecture that there are no bipartite graphs
of defect 4 for {\Delta} \geq 3 and D \geq 5, and comment on some implications
of our results for upper bounds of Nb({\Delta},D).Comment: 25 pages, 14 Postscript figure
On graphs with cyclic defect or excess
The Moore bound constitutes both an upper bound on the order of a graph of
maximum degree and diameter and a lower bound on the order of a graph
of minimum degree and odd girth . Graphs missing or exceeding the
Moore bound by are called {\it graphs with defect or excess
}, respectively.
While {\it Moore graphs} (graphs with ) and graphs with defect or
excess 1 have been characterized almost completely, graphs with defect or
excess 2 represent a wide unexplored area.
Graphs with defect (excess) 2 satisfy the equation
(), where denotes the adjacency matrix of the graph in
question, its order, the matrix whose entries are all
1's, the adjacency matrix of a union of vertex-disjoint cycles, and
a polynomial with integer coefficients such that the matrix
gives the number of paths of length at most joining each pair
of vertices in the graph.
In particular, if is the adjacency matrix of a cycle of order we call
the corresponding graphs \emph{graphs with cyclic defect or excess}; these
graphs are the subject of our attention in this paper.
We prove the non-existence of infinitely many such graphs. As the highlight
of the paper we provide the asymptotic upper bound of
for the number of graphs of odd degree and cyclic defect or excess.
This bound is in fact quite generous, and as a way of illustration, we show the
non-existence of some families of graphs of odd degree and cyclic
defect or excess.
Actually, we conjecture that, apart from the M\"obius ladder on 8 vertices,
no non-trivial graph of any degree and cyclic defect or excess exists.Comment: 20 pages, 3 Postscript figure
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