Related papers: From Integer Sequences to Block Designs via Counti…

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…

Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments…

Walks in a directed graph can be given a partially ordered structure that extends to possibly unconnected objects, called hikes. Studying the incidence algebra on this poset reveals unsuspected relations between walks and self-avoiding…

Suppose that $[n]=\left\{0,1,2,...,n\right\}$ is a set of non-negative integers and $h,k \in [n]$. The $L(h,k)$-labeling of graph $G$ is the function $l:V(G)\rightarrow[n]$ such that $\left|l(u)-l(v)\right|\geq h$ if the distance $d(u,v)$…

Quantum random walks represent a powerful tool for the implementation of various quantum algorithms. We consider a convolution problem for the graphs which provide quantum and classical random walks. We suggest a new method for lattices and…

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block…

In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such…

Let H(n) be the group of 3x3 uni-uppertriangular matrices with entries in Z/nZ, the integers mod n. We show that the simple random walk converges to the uniform distribution in order n^2 steps. The argument uses Fourier analysis and is…

We construct strongly walk-regular graphs as coset graphs of the duals of codes with three non-zero homogeneous weights over $\mathbb{Z}_{p^m},$ for $p$ a prime, and more generally over chain rings of depth $m$, and with a residue field of…

Datasets from several domains, such as life-sciences, semantic web, machine learning, natural language processing, etc. are naturally structured as acyclic graphs. These datasets, particularly those in bio-informatics and computational…

Using spectral graph theory, we show how to obtain inequalities for the number of walks in graphs from nonnegative polynomials and present a new family of such inequalities.

Let $G$ be a finite simple graph with edge ideal $I(G)$. Let $J(G)$ denote the Alexander dual of $I(G)$. We show that a description of all induced cycles of odd length in $G$ is encoded in the associated primes of $J(G)^2$. This result…

We study sequences of random numbers {Z[1],Z[2],Z[3],...,Z[n]} -- which can be considered random walks with reflecting barriers -- and define their "types" according to whether Z[i] > Z[i+1], (a down-movement), or Z[i] < Z[i+1]…

Researchers developing implementations of distributed graph analytic algorithms require graph generators that yield graphs sharing the challenging characteristics of real-world graphs (small-world, scale-free, heavy-tailed degree…

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these…

In this paper we define and study a new family of graphs that generalises the notions of line graphs and path graphs. Let $G$ be a graph with no loops but possibly with parallel edges. An \emph{$\ell$-link} of $G$ is a walk of $G$ of length…

We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by…

Graphs, and sequences of growing graphs, can be used to specify the architecture of mathematical models in many fields including machine learning and computational science. Here we define structured graph "lineages" (ordered by level…

Walks on Young's lattice of integer partitions encode many objects of algebraic and combinatorial interest. Chen et al. established connections between such walks and arc diagrams. We show that walks that start at $\varnothing$, end at a…

Simultaneous Geometric Embedding (SGE) asks whether, for a given collection of graphs on the same vertex set V, there is an embedding of V in the plane that admits a crossing-free drawing with straightline edges for each of the given…