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An edge-colored graph $G$ is $k$-color connected if, between each pair of vertices, there exists a path using at least $k$ different colors. The $k$-color connection number of $G$, denoted by $cc_{k}(G)$, is the minimum number of colors…
A graph is $k$-connected if it has $k$ internally-disjoint paths between every pair of nodes. A subset $S$ of nodes in a graph $G$ is a $k$-connected set if the subgraph $G[S]$ induced by $S$ is $k$-connected; $S$ is an $m$-dominating set…
Vertex connectivity is a well-studied concept in graph theory with numerous applications. A graph is $k$-connected if it remains connected after removing any $k-1$ vertices. The vertex connectivity of a graph is the maximum $k$ such that…
A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice X. Our characterization…
In most studies, street networks are considered as undirected graphs while one-way streets and their effect on shortest paths are usually ignored. Here, we first study the empirical effect of one-way streets in about $140$ cities in the…
A $k$-regular graph is called a divisible design graph if its vertex set can be partitioned into $m$ classes of size $n$, such that two distinct vertices from the same class have exactly $\lambda_1$ common neighbours, and two vertices from…
An $i$-packing in a graph $G$ is a set of vertices at pairwise distance greater than $i$. For a nondecreasing sequence of integers $S=(s\_{1},s\_{2},\ldots)$, the $S$-packing chromatic number of a graph $G$ is the least integer $k$ such…
Suppose $D = (V, E)$ is a strongly connected digraph and $u, v \in V (D)$. Among the many metrics in graphs, the sum metric warrants further exploration. The sum distance $sd(u, v)$ defined as $sd(u, v) =\overrightarrow{d}(u,…
Map vertices of a graph to (not necessarily distinct) points of the plane so that two adjacent vertices are mapped at least a unit distance apart. The plane-width of a graph is the minimum diameter of the image of the vertex set over all…
A mixed graph can be seen as a type of digraph containing some edges (two opposite arcs). Here we introduce the concept of sequence mixed graphs, which is a generalization of both sequence graphs and iterated line digraphs. These structures…
A cactus is a connected graph in which each edge is contained in at most one cycle. We generalize the concept of cactus graphs, i.e., a $k$-cactus is a connected graph in which each edge is contained in at most $k$ cycles where $k\ge 1$. It…
We introduce a new class of problems lying halfway between questions about graph capacity and intersection. We say that two binary sequences x and y of the same length have a skewincidence if there is a coordinate i for which x_i=y_{i+1}=1…
The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex…
We give a complete description of the distance relation on the graph of $4$-ary simplex codes of dimension $2$. This is a connected graph of diameter $3$. For every vertex we determine the sets of all vertices at distance $i\in\{1,2,3\}$…
A circular Pascal array is a periodization of the familiar Pascal's triangle. Using simple operators defined on periodic sequences, we find a direct relationship between the ranges of the circular Pascal arrays and numbers of certain…
We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right…
Betweenness centrality of a vertex in a graph measures the fraction of shortest paths going through the vertex. This is a basic notion for determining the importance of a vertex in a network. The k-betweenness centrality of a vertex is…
A graph is k-linked if any k disjoint vertex-pairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7-, 10- and 13-dimensional…
We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula $4^{n-1}$. These give relations between different statistics, such as the…
We present an exact formula for the ordinary generating series of the simple paths between any two vertices of a graph. Our formula involves the adjacency matrix of the connected induced subgraphs and remains valid on weighted and directed…