Related papers: Regular graphs to induce even periodic Grover walk…
We introduce a new class of transitive permutation groups which properly contains the automorphism groups of vertex-transitive graphs and digraphs. We then give a sufficient condition for a quotient of this family to remain in the family,…
A recent result by Kardo\v{s}, M\'a\v{c}ajov\'a and Zerafa [J. Comb. Theory, Ser. B. 160 (2023) 1--14] related to the famous Berge-Fulkerson conjecture implies that given an arbitrary set of odd pairwise edge-disjoint cycles, say $\mathcal…
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll k \lesssim \log |G|$. The results of this article supplement those in the three main papers on random Cayley…
We make progress on three long standing conjectures from the 1960s about path and cycle decompositions of graphs. Gallai conjectured that any connected graph on $n$ vertices can be decomposed into at most $\left\lceil…
We present a generalized definition of discrete-time quantum walks convenient for capturing a rather broad spectrum of walker's behavior on arbitrary graphs. It includes and covers both: the geometry of possible walker's positions with…
We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of $\tilde{O}(\sqrt{n})$ for graphs on $n$ vertices. The constant-time…
Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to…
Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a…
The lackadaisical quantum walk is a quantum analogue of the lazy random walk obtained by adding a self-loop to each vertex in the graph. We analytically prove that lackadaisical quantum walks can find a unique marked vertex on any regular…
The Kneser graph $K(n, k)$ has as vertices all $k$-element subsets of $[n]=\{1,2,...,n \}$ and an edge between any two vertices that are disjoint. If $n=2k+1$, then $K(n, k)$ is called an odd graph. Let $ n >4$ and $1< k < \frac{n}{2} $. In…
We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly $\ell$-walk-regular with $\ell >1$ if the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two…
We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically…
The continuous-time quantum walk is a particle evolving by Schr\"odinger's equation in discrete space. Encoding the space as a graph of vertices and edges, the Hamiltonian is proportional to the discrete Laplacian. In some physical systems,…
An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield…
In this paper we introduce a novel polynomial-time algorithm to compute graph invariants based on the modified random walk idea on graphs. However not proved to be a full graph invariant by now, our method gives the right answer for the…
Squaring and adding $\pm 1$ mod p generates a curiously intractable random walk. A similar process over the finite field $\mathbf{F}_q$ (with $q=2^d$) leads to novel connections between elementary Galois theory and probability.
We define numbers of the type Oj(N) and Ej(N) and the corresponding integer sequences. We prove that these integer sequences, e.g., SO(N) and SE(N) correspond to the number of odd and even walks in complete graphs. We then prove that there…
We study the existence of periodic colorings and orientations in locally finite graphs. A coloring or orientation of a graph $G$ is periodic if the resulting colored or oriented graph is quasi-transitive, meaning that $V(G)$ has finitely…
Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of…
Let $n\ge 1$ be an integer and $\mathbb{F}_q$ be a finite field of characteristic $p$ with $q$ elements. In this paper, it is proved that the Wenger graph $W_n(q)$ and linearized Wenger graph $L_m(q)$ are edge-girth-regular…