35,801 research outputs found
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 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
Maxima of the Q-index: forbidden 4-cycle and 5-cycle
This paper gives tight upper bounds on the largest eigenvalue q(G) of the
signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let
F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1}
with an edge hanged to its center. It is shown that if G is a graph of order n,
with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,k} be the join of
a complete graph of order k and an independent set of order n-k. It is shown
that if G is a graph of order n, with no 5-cycle, then q(G)<q(S_{n,2}), unless
G=S_{n,k}. It is shown that these results are significant in spectral extremal
graph problems. Two conjectures are formulated for the maximum q(G) of graphs
with forbidden cycles.Comment: 12 page
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