In this lecture, we start with some basic definitions and notation, and then discuss some motivating applications of expanders. A graph G = (V,E) has vertex set V , with |V | = N , and edge set E. G is undirected, unless specified otherwise, and can be a multigraph. A bipartite graph will be denoted as G = (L∪R,E), i.e., its vertex sets are partitioned as L and R (L and R do not necessarily hav...

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family (BDL, 0 < L < ∞) of random metric spaces homeomorphic to the closed unit disk of R2, the space BDL being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangu...

The graph G of order n is an L-borderenergetic which means it has the same Laplacian energy as complete Kn. In this paper, we find that combination bipartite graphs and stars can construct infinite numbers classes graphs. We give two Lborderenergetic under operators union, join their mixed. This research could provide experience for further study structural characteristics

In this lecture, we will construct constant degree D vertex-expanders with expansion of (1 − ε)D (a.k.a. constant degree lossless expanders). This beautiful construction is due to Capalbo-ReingoldVadhan-Wigderson. Concretely, for every constant ε > 0 and every N , we will construct a bipartite graph (L,R,E), |L| = N , |R| = M = poly(ε)N), with left degree D = poly(1ε ), such that every subset S...

We consider log-convex sequences that satisfy an additional constraint imposed on their rate of growth. We call such sequences log-balanced. It is shown that all such sequences satisfy a pair of double inequalities. Sufficient conditions for log-balancedness are given for the case when the sequence satisfies a two(or more-) term linear recurrence. It is shown that many combinatorially interesti...

It has been observed that mutualistic bipartite networks have a nested structure of interactions. In addition, the degree distributions associated with the two guilds involved in such networks (e.g., plants and pollinators or plants and seed dispersers) approximately follow a truncated power law (TPL). We show that nestedness and TPL distributions are intimately linked, and that any biological ...

We consider a model for diffusive phase transitions, for instance, the component separation in a binary mixture. Our model is described by two functions, the absolutete temperature θ := θ(t, x) and the order parameter w := w(t, x), which are governed by a system of two nonlinear parabolic PDEs. The order parameter w is constrained to have double obstacles σ∗ ≤ w ≤ σ∗ (i.e., σ∗ and σ∗ are the th...

We discuss N = 2 SU(2) Yang-Mills gauge theories coupled with Nf (= 2, 3) massive hypermultiplets in the weak coupling limit. We determine the exact massive prepotentials and the monodromy matrices around the weak coupling limit. We also study that the double scaling limit of these massive theories and find that the massive Nf − 1 theory can be obtained from the massive Nf theory. New formulae ...

Abstract For any simple connected graph G of order n , having eigen spectrum μ 1 ≥ 2 ⋯ with middle eigenvalues H and L where = ⌊( + 1)/2⌋ ⌈( 1)/2⌉, the HOMO–LUMO gap is defined as Δ . In this article, a upper bound for corresponding to special class bipartite graphs estimated. As an application, bounds certain classes nanotubes nanotori are