3 research outputs found
Observations on the Lov\'asz -Function, Graph Capacity, Eigenvalues, and Strong Products
This paper provides new observations on the Lov\'{a}sz -function of
graphs. These include a simple closed-form expression of that function for all
strongly regular graphs, together with upper and lower bounds on that function
for all regular graphs. These bounds are expressed in terms of the
second-largest and smallest eigenvalues of the adjacency matrix of the regular
graph, together with sufficient conditions for equalities (the upper bound is
due to Lov\'{a}sz, followed by a new sufficient condition for its tightness).
These results are shown to be useful in many ways, leading to the determination
of the exact value of the Shannon capacity of various graphs, eigenvalue
inequalities, and bounds on the clique and chromatic numbers of graphs. Since
the Lov\'{a}sz -function factorizes for the strong product of graphs,
the results are also particularly useful for parameters of strong products or
strong powers of graphs. Bounds on the smallest and second-largest eigenvalues
of strong products of regular graphs are consequently derived, expressed as
functions of the Lov\'{a}sz -function (or the smallest eigenvalue) of
each factor. The resulting lower bound on the second-largest eigenvalue of a
-fold strong power of a regular graph is compared to the Alon--Boppana
bound; under a certain condition, the new bound is superior in its exponential
growth rate (in ). Lower bounds on the chromatic number of strong products
of graphs are expressed in terms of the order and the Lov\'{a}sz
-function of each factor. The utility of these bounds is exemplified,
leading in some cases to an exact determination of the chromatic numbers of
strong products or strong powers of graphs. The present research paper is aimed
to have tutorial value as well.Comment: electronic links to references were added in version 2; Available at
https://www.mdpi.com/1099-4300/25/1/10
Strong products of Kneser graphs
AbstractLet G ⊠H be the strong product of graphs G and H. We give a short proof that χ(G ⊠H) ⩾χ(G)+2ω(H)−2. Kneser graphs are then used to demonstrate that this lower bound is sharp. We also prove that for every n⩾2 there is an infinite sequence of pairs of graphs G and G′ such that G′ is not a retract of G while G′ ⊠Kn is a retract of G ⊠Kn