Related papers: Isogeny graphs with maximal real multiplication
A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and…
A sequence $\sigma$ of $p$ non-negative integers is unigraphic if it is the degree sequence of exactly one graph, up to isomorphism. A polyhedral graph is a $3$-connected, planar graph. We investigate which sequences are unigraphic with…
Let $\ell \geqslant 0$ be an integer, and $G$ be a graph without loops. An $\ell$-link of $G$ is a walk of length $\ell$ in which consecutive edges are different. We identify an $\ell$-link with its reverse sequence. The $\ell$-link graph…
A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the $2$-torsion subgroup of the domain. A Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is…
Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge intersects itself. We analyze several characteristics of simple drawings of complete…
Consider an algebraic identity between elliptic modular graphs where several vertices are at fixed locations (and hence unintegrated) while the others are integrated over the toroidal worldsheet. At any unintegrated vertex, we can glue an…
Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph…
A simple topological graph T = (V(T), E(T)) is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs G and H…
In this monography, it is proposed to consider the concepts of spectra of edge cuts and edge cycles of a graph as a basic mathematical structure for solving the problem of graph isomorphism. An edge cut is defined by an edge and the…
A set $S$ of isometric paths of a graph $G$ is ``$v$-rooted'', where $v$ is a vertex of $G$, if $v$ is one of the endpoints of all the isometric paths in $S$. The isometric path complexity of a graph $G$, denoted by $ipco{G}$, is the…
The combinatorial refinement techniques have proven to be an efficient approach to isomorphism testing for particular classes of graphs. If the number of refinement rounds is small, this puts the corresponding isomorphism problem in a…
We introduce a class of algebras over a field $\mathbb{F}$ related to directed graphs in which all edges are labeled by nonzero elements of the field $\mathbb{F}$. If all labels are different from $1$, these algebras are axial algebras. We…
An explicit algorithm is presented for testing whether two non-directed graphs are isomorphic or not. It is shown that for a graph of n vertices, the number of n independent operations needed for the test is polynomial in n. A proof that…
The power graph and the enhanced power graph of a group $\mathbf G$ are simple graphs with vertex set $G$; two elements of $G$ are adjacent in the power graph if one of them is a power of the other, and they are adjacent in the enhanced…
We give an algorithm to calculate the minimal and maximal genus of the orientable closed surface where a graph $G$ can be embedded. For this, we construct some special branched coverings of the 2-sphere. We apply this algorithm to calculate…
Dimension 4 isogenies have first been introduced in cryptography for the cryptanalysis of Supersingular Isogeny Diffie-Hellman (SIDH) and have been used constructively in several schemes, including SQIsignHD, a derivative of SQIsign isogeny…
We call a finite undirected graph minimally k-matchable if it has at least k distinct perfect matchings but deleting any edge results in a graph which has not. An odd subdivision of some graph G is any graph obtained by replacing every edge…
Graph-level representations are crucial tools for characterising structural differences between graphs. However, comparing graphs with different cardinalities, even when sampled from the same underlying distribution, remains challenging.…
Mixed graphs can be seen as digraphs that have both arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case in which such graphs are Cayley graphs of Abelian groups. These groups can be constructed by…
A connected graph can be associated with two distinct evolution algebras. In the first case, the structural matrix is the adjacency matrix of the graph itself. In the second case, the structural matrix is the transition probabilities matrix…