2,677 research outputs found
On the structure of graphs without short cycles
The objective of this thesis is to study cages, constructions and properties of such families of graphs. For this, the study of graphs without short cycles plays a fundamental role in order to develop some knowledge on their structure, so we can later deal with the problems on cages. Cages were introduced by Tutte in 1947. In 1963, Erdös and Sachs proved that (k, g) -cages exist for any given values of k and g. Since then, large amount of research in cages has been devoted to their construction.
In this work we study structural properties such as the connectivity, diameter, and degree regularity of graphs without short cycles.
In some sense, connectivity is a measure of the reliability of a network. Two graphs with the same edge-connectivity, may be considered to have different reliabilities, as a more refined index than the edge-connectivity, edge-superconnectivity is proposed together with some other parameters called restricted connectivities.
By relaxing the conditions that are imposed for the graphs to be cages, we can achieve more refined connectivity properties on these families and also we have an approach to structural properties of the family of graphs with more restrictions (i.e., the cages).
Our aim, by studying such structural properties of cages is to get a deeper insight into their structure so we can attack the problem of their construction.
By way of example, we studied a condition on the diameter in relation to the girth pair of a graph, and as a corollary we obtained a result guaranteeing restricted connectivity of a special family of graphs arising from geometry, such as polarity graphs.
Also, we obtained a result proving the edge superconnectivity of semiregular cages. Based on these studies it was possible to develop the study of cages.
Therefore obtaining a relevant result with respect to the connectivity of cages, that is, cages are k/2-connected. And also arising from the previous work on girth pairs we obtained constructions for girth pair cages that proves a bound conjectured by Harary and Kovács, relating the order of girth pair cages with the one for cages. Concerning the degree and the diameter, there is the concept of a Moore graph, it was introduced by Hoffman and Singleton after Edward F. Moore, who posed the question of describing and classifying these graphs.
As well as having the maximum possible number of vertices for a given combination of degree and diameter, Moore graphs have the minimum possible number of vertices for a regular graph with given degree and girth. That is, any Moore graph is a cage. The formula for the number of vertices in a Moore graph can be generalized to allow a definition of Moore graphs with even girth (bipartite Moore graphs) as well as odd girth, and again these graphs are cages. Thus, Moore graphs give a lower bound for the order of cages, but they are known to exist only for very specific values of k, therefore it is interesting to study how far a cage is from this bound, this value is called the excess of a cage.
We studied the excess of graphs and give a contribution, in the sense of the work of Biggs and Ito, relating the bipartition of girth 6 cages with their orders. Entire families of cages can be obtained from finite geometries, for example, the graphs of incidence of projective planes of order q a prime power, are (q+1, 6)-cages. Also by using other incidence structures such as the generalized quadrangles or generalized hexagons, it can be obtained families of cages of girths 8 and 12.
In this thesis, we present a construction of an entire family of girth 7 cages that arises from some combinatorial properties of the incidence graphs of generalized quadrangles of order (q,q)
Bounds for identifying codes in terms of degree parameters
An identifying code is a subset of vertices of a graph such that each vertex
is uniquely determined by its neighbourhood within the identifying code. If
\M(G) denotes the minimum size of an identifying code of a graph , it was
conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there
exists a constant such that if a connected graph with vertices and
maximum degree admits an identifying code, then \M(G)\leq
n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any ,
\M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs
containing, among others, all regular graphs and all graphs of bounded clique
number. This settles the conjecture (up to constants) for these classes of
graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}.
In a second part, we prove that in any graph of minimum degree and
girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n.
Using the former result, we give sharp estimates for the size of the minimum
identifying code of random -regular graphs, which is about
Compressed Sensing Using Binary Matrices of Nearly Optimal Dimensions
In this paper, we study the problem of compressed sensing using binary
measurement matrices and -norm minimization (basis pursuit) as the
recovery algorithm. We derive new upper and lower bounds on the number of
measurements to achieve robust sparse recovery with binary matrices. We
establish sufficient conditions for a column-regular binary matrix to satisfy
the robust null space property (RNSP) and show that the associated sufficient
conditions % sparsity bounds for robust sparse recovery obtained using the RNSP
are better by a factor of compared to the
sufficient conditions obtained using the restricted isometry property (RIP).
Next we derive universal \textit{lower} bounds on the number of measurements
that any binary matrix needs to have in order to satisfy the weaker sufficient
condition based on the RNSP and show that bipartite graphs of girth six are
optimal. Then we display two classes of binary matrices, namely parity check
matrices of array codes and Euler squares, which have girth six and are nearly
optimal in the sense of almost satisfying the lower bound. In principle,
randomly generated Gaussian measurement matrices are "order-optimal". So we
compare the phase transition behavior of the basis pursuit formulation using
binary array codes and Gaussian matrices and show that (i) there is essentially
no difference between the phase transition boundaries in the two cases and (ii)
the CPU time of basis pursuit with binary matrices is hundreds of times faster
than with Gaussian matrices and the storage requirements are less. Therefore it
is suggested that binary matrices are a viable alternative to Gaussian matrices
for compressed sensing using basis pursuit. \end{abstract}Comment: 28 pages, 3 figures, 5 table
Generalized Tur\'an problems for even cycles
Given a graph and a set of graphs , let
denote the maximum possible number of copies of in an -free
graph on vertices. We investigate the function , when
and members of are cycles. Let denote the cycle of
length and let . Some of our main
results are the following.
(i) We show that for any .
Moreover, we determine it asymptotically in the following cases: We show that
and that the maximum
possible number of 's in a -free bipartite graph is .
(ii) Solymosi and Wong proved that if Erd\H{o}s's Girth Conjecture holds,
then for any we have . We prove that forbidding any other even cycle
decreases the number of 's significantly: For any , we have
More generally,
we show that for any and such that , we have
(iii) We prove provided a
strong version of Erd\H{o}s's Girth Conjecture holds (which is known to be true
when ). Moreover, forbidding one more cycle decreases the number
of 's significantly: More precisely, we have and for .
(iv) We also study the maximum number of paths of given length in a
-free graph, and prove asymptotically sharp bounds in some cases.Comment: 37 Pages; Substantially revised, contains several new results.
Mistakes corrected based on the suggestions of a refere
Balanced Allocation on Graphs: A Random Walk Approach
In this paper we propose algorithms for allocating sequential balls into
bins that are interconnected as a -regular -vertex graph , where
can be any integer.Let be a given positive integer. In each round
, , ball picks a node of uniformly at random and
performs a non-backtracking random walk of length from the chosen node.Then
it allocates itself on one of the visited nodes with minimum load (ties are
broken uniformly at random). Suppose that has a sufficiently large girth
and . Then we establish an upper bound for the maximum number
of balls at any bin after allocating balls by the algorithm, called {\it
maximum load}, in terms of with high probability. We also show that the
upper bound is at most an factor above the lower bound that is
proved for the algorithm. In particular, we show that if we set , for every constant , and
has girth at least , then the maximum load attained by the
algorithm is bounded by with high probability.Finally, we
slightly modify the algorithm to have similar results for balanced allocation
on -regular graph with and sufficiently large girth
- …