A graph G is 1-Hamilton-connected if G − x is Hamilton-connected for every vertex x ∈ V (G). In the paper we introduce a closure concept for 1-Hamiltonconnectedness in claw-free graphs. If G is a (new) closure of a claw-free graph G, then G is 1-Hamilton-connected if and only if G is 1-Hamilton-connected, G is the line graph of a multigraph, and for some x ∈ V (G), G − x is the line graph of a ...

let g be a connected simple (molecular) graph. the distance d(u, v) between two vertices u and v of g is equal to the length of a shortest path that connects u and v. in this paper we compute some distance based topological indices of h-phenylenic nanotorus. at first we obtain an exactformula for the wiener index. as application we calculate the schultz index and modified schultz index of this ...

Let A(n; k; t) denote the smallest integer e for which every k-connected graph on n vertices can be made (k + t)-connected by adding e new edges. We determine A(n; k; t) for all values of n; k and t in the case of (directed and undirected) edge-connectivity and also for directed vertex-connectivity. For undirected vertex-connectivity we determine A(n; k; 1) for all values of n and k. We also de...

A graph is textit{symmetric}, if its automorphism group is transitive on the set of its arcs. In this paper, we  classifyall the connected cubic symmetric  graphs of order $36p$  and $36p^{2}$, for each prime $p$, of which the proof depends on the classification of finite simple groups.

For any simple connected undirected graph, it is well known that the Kirchhoff and multiplicative degree-Kirchhoff indices can be computed using the Laplacian matrix. We show that the same is true for the additive degree-Kirchhoff index and give a compact Matlab program that computes all three Kirchhoffian indices with the Laplacian matrix as the only input.

 The order graph of a group $G$, denoted by $Gamma^*(G)$, is a graph whose vertices are subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $|H|big{|}|K|$ or $|K|big{|}|H|$. In this paper, we study the connectivity and diameter of this  graph. Also we give a relation between the order graph and prime  graph of a group.

The relationship between graph coloring and the immersion order is considered. Vertex connectivity, edge connectivity and related issues are explored. It is shown that a t-chromatic graph G contains either an immersed Kt or an immersed t-chromatic subgraph that is both 4-vertex-connected and t-edge-connected. This gives supporting evidence of our conjecture that if G requires at least t colors,...

A subset T ⊆ V of terminals is k-connected to a root s in a directed/undirected graph J if J has k internally-disjoint vs-paths for every v ∈ T ; T is k-connected in J if T is k-connected to every s ∈ T . We consider the Subset k-Connectivity Augmentation problem: given a graph G = (V,E) with edge/node-costs, node subset T ⊆ V , and a subgraph J = (V,EJ ) of G such that T is k-connected in J , ...

Let (X,R) be a commutative association scheme and let Γ = (X,R ∪R>) be a connected undirected graph where R ∈ R. Godsil (resp., Brouwer) conjectured that the edge connectivity (resp., vertex connectivity) of Γ is equal to its valency. In this paper, we prove that the deletion of the neighborhood of any vertex leaves behind at most one non-singleton component. Two distinct vertices a, b ∈ X are ...