Related papers: Classification of 3-GNDB Graphs

Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$…

A simple and connected $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$, such that adjacent vertices have relatively prime labels. We will present previously…

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple $(v,u,x,y)$ of vertices such that both $(v,u,x)$ and $(u,x,y)$ are paths of length two. The 3-arc graph of a graph $G$ is defined to have vertices the arcs of $G$ such that two…

A connected graph is called of non-QE class if it does not admit a quadratic embedding in a Euclidean space. A non-QE graph is called primary if it does not contain a non-QE graph as an isometrically embedded proper subgraph. The graphs on…

A finite non-increasing sequence of positive integers $d = (d_1\geq \cdots\geq d_n)$ is called a degree sequence if there is a graph $G = (V,E)$ with $V = \{v_1,\ldots,v_n\}$ and $deg(v_i)=d_i$ for $i=1,\ldots,n$. In that case we say that…

A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of positive edges. In the adjacency matrix of a signed graph, entries can be $0$,…

Aims. An attempt to classify gamma-ray bursts (GRBs) with a low level of supervision using the state-of-the-start approaches stemming from graph theory was undertaken. Methods. Graph-based classification methods, relying on different…

A connected graph $\Gamma$ of diameter ${\rm diam}(\Gamma) \ge \ell$ is $\ell$-distance-balanced if $|W_{xy}(\Gamma)|=|W_{yx}(\Gamma)|$ for every $x,y\in V(\Gamma)$ with $d_{\Gamma}(x,y)=\ell$, where $W_{xy}(\Gamma)$ is the set of vertices…

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and…

The intersection graph of a group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$…

A non-complete \drg $\Gamma$ is called geometric if there exists a set $\mathcal{C}$ of Delsarte cliques such that each edge of $\Gamma$ lies in a unique clique in $\mathcal{C}$. In this paper, we determine the non-complete distance-regular…

A novel class of boundary conditions is introduced as a generalization of the previously defined class of soft-and-hard/DB (SHDB) boundary conditions. It is shown that the conditions for the generalized soft-and-hard/DB (GSHDB) boundary…

A convex geometric graph is a graph whose vertices are the corners of a convex polygon P in the plane and whose edges are boundary edges and diagonals of the polygon. It is called triangulation-free if its non-boundary edges do not contain…

We classify the ultrahomogeneous complete 3-edge-coloured graphs (3-graphs) with simple theory. This extends Lachlan's result (a corollary of the Effective Classification Theorem for stable structures) classifying the stable homogeneous…

A graph is called integral if all its eigenvalues are integers. A Cayley graph is called normal if its connection set is a union of conjugacy classes. We show that a non-empty integral normal Cayley graph for a group of odd order has an odd…

For a simple graph $G$, the $2$-distance graph, $D_2(G)$, is a graph with the vertex set $V(G)$ and two vertices are adjacent if and only if their distance is $2$ in the graph $G$. In this paper, we characterize all graphs with connected…

A weighted digraph is balanced if the sums of the weights of the incoming and of the outgoing edges are equal at each vertex. We show that if these sums are integers, then the edge weights can be integers as well.

A consistent path system in a graph $G$ is an intersection-closed collection of paths, with exactly one path between any two vertices in $G$. We call $G$ metrizable if every consistent path system in it is the system of geodesic paths…

An independent set in a graph G is a set of vertices no two of which are joined by an edge. A vertex-weighted graph associates a weight with every vertex in the graph. A vertex-weighted graph G is called a unique independence…

A graph $G$ is called well-covered if all maximal independent sets of vertices have the same cardinality. A well-covered graph $G$ is called uniformly well-covered if there is a partition of the set of vertices of $G$ such that each maximal…