Related papers: Group Connectivity: $\mathbb Z_4$ v. $\mathbb Z_2^…
The power graph of a group is the simple graph with vertices as the group elements, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. We study (minimal) cut-sets of the…
We estimate the minimum number of vertices of a cubic graph with given oddness and cyclic connectivity. We prove that a bridgeless cubic graph $G$ with oddness $\omega(G)$ other than the Petersen graph has at least $5.41\cdot\omega(G)$…
The double graph of a graph $G$ is defined as $\mathcal{D}[G]$ = $G \times T_2$, where \(T_2\) is the total graph with 2 vertices and $\times$ stands for the Kronecker product of graphs. In this paper, sufficient conditions for double…
Let the class A of graphs be bridge-addable; that is, whenever a graph G in A has vertices u and v in different components then the graph G+uv is in A. For a random graph sampled uniformly from the graphs in A on vertex set {1,..,n}, there…
The vertex connectivity of a graph $G$ is the size of the smallest set of vertices $S$ such that $G \setminus S$ is disconnected. For the class of planar graphs, the problem of vertex connectivity is well-studied, both from structural and…
Graph connectivity is a fundamental combinatorial optimization problem that arises in many practical applications, where usually a spanning subgraph of a network is used for its operation. However, in the real world, links may fail…
The graph braid group of a complete bipartite graph is the fundamental group of a configuration space of points on the graph, which is a CAT(0) cube complex. We combine an analysis of the topology of links of vertices in this complex, the…
Random key graphs represent topologies of secure wireless sensor networks that apply the seminal Eschenauer-Gligor random key predistribution scheme to secure communication between sensors. These graphs have received much attention and also…
A graph $G$ admiting a $2$-factor is \textit{pseudo $2$-factor isomorphic} if the parity of the number of cycles in all its $2$-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo $2$-factor…
A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that…
We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families,…
Let $G$ be a graph of order $n$ and let $u,v$ be vertices of $G$. Let $\kappa_G(u,v)$ denote the maximum number of internally disjoint $u$-$v$ paths in $G$. Then the average connectivity $\overline{\kappa}(G)$ of $G$, is defined as $…
For $k \ge 4$, let $Q_{2k}$ and $V_{2k}$ denote the ladder and M\"obius ladder on $2k$ vertices, respectively. We prove results that build on a result by Wormald that states that any cyclically $4$-connected cubic graph other than $Q_8$ or…
Finding groups of connected individuals in large graphs with tens of thousands or more nodes has received considerable attention in academic research. In this paper, we analyze three main issues with respect to the recent influx of papers…
In recent years, discrete spaces such as graphs attract much attention as models for physical spacetime or as models for testing the spirit of non-commutative geometry. In this work, we construct the differential algebras for graphs by…
The Evasiveness conjecture have been proved for properties of graphs on a prime-power number of vertices and the six vertices case. The ten vertices case is still unsolved. In this paper we study the size of the automorphism group of a…
The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of…
The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between two people depends on their attributes, such as their age, address, and…
The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $\lambda_k(G)$, recently. In this paper we summarize…
Due to the increasing discovery and implementation of networks within all disciplines of life, the study of subgraph connectivity has become increasingly important. Motivated by the idea of community (or sub-graph) detection within a…