Related papers: The lollipop graph is determined by its spectrum

Two vertices of an odd-distance graph are connected by an edge if and only if their Euclidean distance is an odd integer. We construct a 6-chromatic odd-distance graph in the plane.

An edge-colored graph $G$ is called properly colored if every two adjacent edges are assigned different colors. A monochromatic triangle is a cycle of length 3 with all the edges having the same color. Given a tree $T_0$, let…

For natural numbers $k<n$ we study the graphs $T_{n,k}:=K_{k}\lor\overline{K_{n-k}}$. For $k=1$, $T_{n,1}$ is the star $S_{n-1}$. For $k>1$ we refer to $T_{n,k}$ as a \emph{graph of pyramids}. We prove that the graphs of pyramids are…

A graph $G$ is said to be determined by the spectrum of its Laplacian matrix (DLS) if every graph with the same spectrum is isomorphic to $G$. van Dam and Haemers (2003) conjectured that almost all graphs have this property, but that is…

A mixed graph is obtained from a graph by orienting some of its edges. The Hermitian adjacency matrix of a mixed graph with the vertex set $ \{v_{1}, \ldots , v_{n}\} $, is the matrix $ H=[h_{ij}]_{n \times n} $, where $ h_{ij}=-h_{ji}=i $…

A strong odd coloring of a simple graph $G$ is a proper coloring of the vertices of $G$ such that for every vertex $v$ and every color $c$, either $c$ is used an odd number of times in the open neighborhood $N_G(v)$ or no neighbor of $v$ is…

An odd coloring of a graph $G$ is a proper coloring of $G$ such that for every non-isolated vertex $v$, there is a color appearing an odd number of times in $N_G(v)$. Odd coloring of graphs was studied intensively in recent few years. In…

Inspired by the study of loose cycles in hypergraphs, we define the \emph{loose core} in hypergraphs as a structure which mirrors the close relationship between cycles and $2$-cores in graphs. We prove that in the $r$-uniform binomial…

For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same…

This article deals with homomorphisms of oriented graphs with respect to push equivalence. Here homomorphisms refer to arc preserving vertex mappings, and push equivalence refers to the equivalence class of orientations of a graph $G$ those…

Consider a (not necessarily near-critical) random graph running in continuous time. A recent breadth-first-walk construction is extended in order to account for the surplus edge data in addition to the spanning edge data. Two different…

Silverman and Stange define the notion of an aliquot cycle of length L for a fixed elliptic curve E defined over the rational numbers, and conjecture an order of magnitude for the function which counts such aliquot cycles. In the present…

A Random walk labeling of a graph $G$ is any labeling of $G$ that could have been obtained by performing a random walk on $G$. Continuing two recent works, we calculate the number of random walk labelings of perfect trees, combs, and double…

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $\Delta^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A…

It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every…

We show that if $G$ is a $n$-vertex connected chordal graph, then it admits a longest path transversal of size $O(\log^2 n)$. Under the stronger assumption of 2-connectivity, we show $G$ admits a longest cycle transversal of size $O(\log…

Many applications, ranging from natural to social sciences, rely on graphlet analysis for the intuitive and meaningful characterization of networks employing micro-level structures as building blocks. However, it has not been thoroughly…

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…

We address the Laplacian on a perturbed periodic graph which might not be a periodic graph. We present a class of perturbed graphs for which the essential spectra of the Laplacians are stable even when the graphs are perturbed by adding and…

In a coalescing random walk, a set of particles make independent random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues the random walk through the graph.…