Related papers: Relative complements and a `switch'-classification…
The vertices of a graph are classified into seven types by J.T. Hedetniemi, S.M. Hedetniemi, S.T. Hedetniemi and T.M. Lewis and they ask the following questions: 1) What is the smallest order $n$ of a graph having $n-2$ very typical…
We give combinatorial proofs of some enumeration formulas involving labelled threshold, quasi-threshold, loop-threshold and quasi-loop-threshold graphs. In each case we count by number of vertices and number of components. For threshold…
A graph is apex if it can be made planar by deleting a vertex, that is, $\exists v$ such that $G-v$ is planar. We define the related notions of edge apex, $\exists e$ such that $G-e$ is planar, and contraction apex, $\exists e$ such that…
Switching is an operation on a graph that does not change the spectrum of the adjacency matrix, thus producing cospectral graphs. An important activity in the field of spectral graph theory is the characterization of graphs by their…
We consider the graphs whose edges are marked by the integers (weights) from $0$ to $q-1$ (zero corresponds to no-edge). Such graph is called additive if its vertices can be marked in such a way that the weight of every edge is equal to the…
We characterize the spectrum of the Laplacian of graphs composed of one or two finite or infinite chains connected to a complete graph. We show the existence of localized eigenvectors of two types, eigenvectors that vanish exactly outside…
Let $(X,E_X)$ and $(V,E_V)$ be finite connected graphs without loops. We assume that $V$ has two distinguished vertices $a,b$ and an automorphism $\gamma$ which exchanges $a$ and~$b$. The $V$-edge substitution of $X$ is the graph $X[V]$…
The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix. In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph. By defining two types of…
Local complementation of a graph $G$ on vertex $v$ is an operation that results in a new graph $G*v$, where the neighborhood of $v$ is complemented. Two graph are locally equivalent if on can be reached from the other one through local…
There are local operators on (labeled) graphs $G$ with labels $(g_{ij})$ coming from a finite field. If the filed is binary, in other words, if the graph is ordinary, the operation is just the local complementation. That is, to choose a…
It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…
The number of topologies and non-homeomorphic topologies on a fixed finite set are now known up to $n=18$, $n=16$ but still no complete formula yet (Sloane). There are one to one correspondence among topologies, preorder and digraphs. In…
We find all maximal linklessly embeddable graphs of order up to 11, and verify that for every graph $G$ of order 11 either $G$ or its complement $cG$ is intrinsically linked. We give an example of a graph $G$ of order 11 such that both $G$…
Inspired by applications of perfect graphs in combinatorial optimization, Chv\'{a}tal defined t-perfect graphs in 1970s. The long efforts of characterizing t-perfect graphs started immediately, but embarrassingly, even a working conjecture…
We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs…
The simplex graph $S(G)$ of a graph $G$ is defined as the graph whose vertices are the cliques of $G$ (including the empty set), with two vertices being adjacent if, as cliques of $G$, they differ in exactly one vertex. Simplex graphs form…
A graph is square-complementary (squco, for short) if its square and complement are isomorphic. We prove that there are no squco graphs with girth 6, that every bipartite graph is an induced subgraph of a squco bipartite graph, that the…
We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks.…
Suppose $G$ is a undirected simple graph. A $k-$subset of edges in $G$ without common vertices is called a $k-$matching and the number of such subsets is denoted by $p(G,k)$. The aim of this paper is to present exact formulas for $p(G,3)$,…
In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency…