Related papers: Symmetric Squares of Graphs
We study a class of symmetric quantum walks on Hamming graphs, where the distance between vertices specifies the transition probability. A special model is the simple quantum walk on the hypercube, which has been discussed in the…
We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently probability transition matrix. This construction can be used to form pairs of cospectral…
Let $S(H)$ be the set of all self-adjoint bonded linear operators on $H$ and $\mathcal{V} \subset S(H)$ a subset that is pertinent in mathematical foundations of quantum mechanics. A symmetry is a bijective map $\phi :\mathcal{V} \to…
The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system, whose hamiltonian is identical to the adjacency matrix of a…
We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges…
Without imposing restrictions on a weighted graph's arc lengths, symmetry structures cannot be expected. But, they exist. To find them, the graphs are decomposed into a component that dictates all closed path properties (e.g., shortest and…
We consider bounds on maximum nullity of a graph via transversal numbers of compatible collections of forts. Results include generalizations of theorems from symmetric to combinatorially symmetric matrices, special bases of matrix…
We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…
We establish upper and lower bounds on the dimension of the space spanned by the symmetric powers of the natural character of generalised symmetric groups. We adapt the methods of Savitt and Stanley from their paper `A note on the symmetric…
Recently normalized Laplacian matrices of graphs are studied as density matrices in quantum mechanics. Separability and entanglement of density matrices are important properties as they determine the nonclassical behavior in quantum…
For non-Hermitian equilateral q-pointed star-shaped quantum graphs of paper I [Can. J. Phys. 90, 1287 (2012), arXiv 1205.5211] we show that due to certain dynamical aspects of the model as controlled by the external, rotation-symmetric…
In light of Lov\'{a}sz's longstanding question on the existence of Hamilton paths in vertex-transitive graphs, this paper considers a natural variant: what if vertex-transitivity is relaxed, yet a high degree of symmetry--specifically…
Strongly regular graphs are highly symmetrical and can be described fully with just a few parameters, yet the existence of many of them is still under the question. In this paper, we continue the study of the famuly of strongly regular…
The positive and negative square energies of a graph, $s^+(G)$ and $s^-(G)$, are the sums of squares of the positive and negative eigenvalues of the adjacency matrix, respectively. The first results on square energies revealed symmetry…
A non-standard generalisation of the Bender potentials $x^2(\ii x^\ve)$ is suggested. The spectra are obtained numerically and some of their particular properties are discussed.
Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the…
A formula for computing the metric dimension of a simple graph, having no singleton twin, is given. A sufficient condition for a simple graph to have the exchange property, for resolving sets, is found. Some families of power graphs of…
We study the properties of certain graphs involving the sums of primes. Their structure largely turns out to relate to the distribution of prime gaps and can be roughly seen in Cram\'er's model as well. We also discuss generalizations to…
According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In…
A natural problem in combinatorial rigidity theory concerns the determination of the rigidity or flexibility of bar-joint frameworks in $\mathbb{R}^d$ that admit some non-trivial symmetry. When $d=2$ there is a large literature on this…