Related papers: Fusions of the generalized Hamming scheme on a str…
An imprimitive symmetric indecomposable association scheme of rank 5 is said to be Higmanian. A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a symmetric divisible design. We establish conditions which…
Given a graph $H$, we investigate the $d$-regular graphs $G$ with the highest $H$-density. We reframe the problem as a continuous optimization problem on the eigenvalues of $G$ by relating injective homomorphism numbers from $H$ and…
Traditional graph centrality measures effectively quantify node importance but fail to capture the structural uniqueness of multi-scale connectivity patterns -- critical for understanding network resilience and function. This paper…
It is shown that if T is a connected nontrivial graph and X is an arbitrary finite simplicial complex, then there is a graph G such that the complex Hom(T,G) is homotopy equivalent to X. The proof is constructive, and uses a nerve lemma.…
A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that every graph is…
We investigate 1-factorisations in which the 2-regular graphs that occur as the union of a pair of 1-factors appear an equal number of times across the unions of all pairs of 1-factors in the 1-factorisation. We call such 1-factorisations…
A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U = \alpha A + \beta I + \gamma J + \delta D$ with $\alpha \neq 0$, where $I$ is the identity matrix, $J$ is the all-ones matrix and $D$ is the…
Random graphs offer a useful mathematical representation of a variety of real world complex networks. Exponential random graphs, for example, are particularly suited towards generating random graphs constrained to have specified statistical…
The $\alpha$-Hermitian adjacency matrix $H_\alpha$ of a mixed graph $X$ has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number…
We look at constructions of aperiodic SFTs on fundamental groups of graph of groups. In particular we prove that all generalized Baumslag-Solitar groups (GBS) admit a strongly aperiodic SFT. Our proof is based on a structural theorem by…
It is shown that a flat subgroup, $H$, of the totally disconnected, locally compact group $G$ decomposes into a finite number of subsemigroups on which the scale function is multiplicative. The image, $P$, of a multiplicative semigroup in…
Given a graph $H$, a graph $G$ is $H$-free if $G$ does not contain $H$ as an induced subgraph. Shi and Shan conjectured that every $1$-tough $2k$-connected $(P_2 \cup kP_1)$-free graph is hamiltonian for $k \geq 4$. This conjecture has been…
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the…
The supersymmetric Poisson Sigma model is studied as a possible worldsheet realization of generalized complex geometry. Generalized complex structures alone do not guarantee non-manifest N=(2,1) or N=(2,2) supersymmetry, but a certain…
Cospectral graphs are a fascinating concept in graph theory, where two non-isomorphic graphs possess identical sets of eigenvalues. In this paper, we compute the $A_\alpha$-characteristic polynomial of neighbour and non-neighbour splitting…
We investigate generalisations of 1-factorisations and hyperfactorisations of the complete graph $K_{2n}$. We show that they are special subsets of the association scheme obtained from the Gelfand pair $(S_{2n},S_2 \wr S_n)$. This unifies…
An association scheme is $P$-polynomial if and only if it consists of the distance matrices of a distance-regular graph. Recently, bivariate $P$-polynomial association schemes of type $(\alpha,\beta)$ were introduced by Bernard et al., and…
We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Holder data. The construction is geometric, relying on the graph of a single variable…
It is known that a graph isomorphism testing algorithm is polynomially equivalent to a detecting of a graph non-trivial automorphism algorithm. The polynomiality of the latter algorithm, is obtained by consideration of symmetry properties…
The aim of this note is to present some new explicit examples of $O(d,d)$-generalised Leibniz parallelisable spaces arising as the normal bundles of adjoint orbits $\mathcal{O}$ of some semi-simple Lie group $G$. Using this construction, an…