108,177 research outputs found
Degree Sequence Index Strategy
We introduce a procedure, called the Degree Sequence Index Strategy (DSI), by
which to bound graph invariants by certain indices in the ordered degree
sequence. As an illustration of the DSI strategy, we show how it can be used to
give new upper and lower bounds on the -independence and the -domination
numbers. These include, among other things, a double generalization of the
annihilation number, a recently introduced upper bound on the independence
number. Next, we use the DSI strategy in conjunction with planarity, to
generalize some results of Caro and Roddity about independence number in planar
graphs. Lastly, for claw-free and -free graphs, we use DSI to
generalize some results of Faudree, Gould, Jacobson, Lesniak and Lindquester
Relative Fractional Independence Number and Its Applications
We define the relative fractional independence number of two graphs, and
, as where the maximum is taken over all graphs , is the
strong product of and , and denotes the independence number. We
give a non-trivial linear program to compute and discuss some
of its properties. We show that where
can be the independence number, the zero-error Shannon capacity, the
fractional independence number, the Lov'{a}sz number, or the Schrijver's or
Szegedy's variants of the Lov'{a}sz number of a graph . This inequality is
the first explicit non-trivial upper bound on the ratio of the invariants of
two arbitrary graphs, as mentioned earlier, which can also be used to obtain
upper or lower bounds for these invariants. As explicit applications, we
present new upper bounds for the ratio of the zero-error Shannon capacity of
two Cayley graphs and compute new lower bounds on the Shannon capacity of
certain Johnson graphs (yielding the exact value of their Haemers number).
Moreover, we show that the relative fractional independence number can be used
to present a stronger version of the well-known No-Homomorphism Lemma. The
No-Homomorphism Lemma is widely used to show the non-existence of a
homomorphism between two graphs and is also used to give an upper bound on the
independence number of a graph. Our extension of the No-Homomorphism Lemma is
computationally more accessible than its original version
On the Signed -independence Number of Graphs
In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area
Graph-theoretical Bounds on the Entangled Value of Non-local Games
We introduce a novel technique to give bounds to the entangled value of
non-local games. The technique is based on a class of graphs used by Cabello,
Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta
number and is efficiently computable; the lower one is based on the quantum
independence number, which is a quantity used in the study of
entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum
Computation, Communication, and Cryptography (TQC 2014
Bounds for the independence number of a graph
The independence number of a graph is the maximum number of vertices from the vertex set of the graph such that no two vertices are adjacent. We systematically examine a collection of upper bounds for the independence number to determine graphs for which each upper bound is better than any other upper bound considered. A similar investigation follows for lower bounds. In several instances a graph cannot be found. We also include graphs for which no bound equals and bounds which do not apply to general graphs
Counting independent sets in cubic graphs of given girth
We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane. We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph. We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.Postprint (author's final draft
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