Related papers: Ramanujan Graphs

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and…

Presented are polynomial identities which imply generalizations of Euler and Rogers--Ramanujan identities. Both sides of the identities can be interpreted as generating functions of certain restricted partitions. We prove the identities by…

Hypergraphs were introduced in 1973 by Berg\'e. This review aims at giving some hints on the main results that we can find in the literature, both on the mathematical side and on their practical usage. Particularly, different definitions of…

A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs,…

In this paper, a survey about recent progress on problems solved using graph amalgamations is presented, along with some new results with complete proofs, and some related open problems.

We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph.

The theory of {\Gamma}-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion. This new approach is then applied to two graph-enumeration problems which were previously unsolved in…

Graphons, short for graph functions, are limiting objects for sequences of large, finite graphs with respect to the so-called cut metric. In this expository piece, we define graphons, motivate them, and discuss how they complete the space…

We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.

Motivated by a fundamental geometrical object, the cut locus, we introduce and study a new combinatorial structure on graphs.

This paper surveys recent development of concepts related to coloring of signed graphs. Various approaches are presented and discussed.

We study the spectra of cyclic signatures of finite graphs and the corresponding cyclic lifts. Starting from a bipartite Ramanujan graph, we prove the existence of an infinite tower of $3$-cyclic lifts, each of which is again Ramanujan.

We use a q-series identity by Ramanujan to give a combinatorial interpretation of Ramanujan's tau function which involves t-cores and a new class of partitions which we call (m,k)-capsids. The same method can be applied in conjunction with…

The vertices of an interval graph represent intervals over a real line where overlapping intervals denote that their corresponding vertices are adjacent. This implies that the vertices are measurable by a metric and there exists a linear…

In this paper, we define irregular bipolar fuzzy graphs and its various classifications. Size of regular bipolar fuzzy graphs is derived. The relation between highly and neighbourly irregular bipolar fuzzy graphs are established. Some basic…

Let $K$ be a number field. This paper considers arithmetic functions over $K$, that are, complex valued functions on the set of nonzero integral ideals in $K$. Firstly we generalize some basic results on arithmetic functions. Next we define…

In an attempt to prove the Graceful Tree Conjecture, we present two propagation of graphs. The first is to propagate graceful graphs, and the second is to propagate trees from a gracefully labeled tree. The motivation in propagating such…

We study topological properties of the graph topology.

Human knowledge provides a formal understanding of the world. Knowledge graphs that represent structural relations between entities have become an increasingly popular research direction towards cognition and human-level intelligence. In…

A new kind of diagrams is presented, showing the causal structure of bimetric interactions.