Gap sets for the spectra of cubic graphs

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals ( 2 2 , 3 ) (2 \sqrt {2},3) and [ − 3 , − 2 ) [-3,-2) achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [ − 3 , 3 ] [-3,3] which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in [ − 3 , 3 ) [-3,3) can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

New and explicit constructions of unbalanced Ramanujan bipartite graphs

The objectives of this article are threefold. Firstly, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implementing the various construction methods. The third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of “prohibited” edges? We provide an affirmative answer in many cases, as long as the set of prohibited edges is not too large.

Ramanujan Graphs for Post-Quantum Cryptography

We introduce a cryptographic hash function based on expander graphs, suggested by Charles et al. ’09, as one prominent candidate in post-quantum cryptography. We propose a generalized version of explicit constructions of Ramanujan graphs, which are seen as an optimal structure of expander graphs in a spectral sense, from the previous works of Lubotzky, Phillips, Sarnak ’88 and Chiu ’92. We also describe the relationship between the security of Cayley hash functions and word problems for group theory. We also give a brief comparison of LPS-type graphs and Pizer’s graphs to draw attention to the underlying hard problems in cryptography.

Generalized Group–Subgroup Pair Graphs

A regular finite graph is called a Ramanujan graph if its zeta function satisfies an analog of the Riemann Hypothesis. Such a graph has a small second eigenvalue so that it is used to construct cryptographic hash functions. Typically, explicit family of Ramanujan graphs are constructed by using Cayley graphs. In the paper, we introduce a generalization of Cayley graphs called generalized group–subgroup pair graphs, which are a generalization of group–subgroup pair graphs defined by Reyes-Bustos. We study basic properties, especially spectra of them.