general formulas are obtained for the vertex padmakar-ivan index (piv) of tetrathiafulvalene(ttf) dendrimer, whereby ttf units we are employed as branching centers. the piv index isa wiener-szeged-like index developed very recently. this topological index is defined as thesummation of all sums of nu(e) and nv(e), over all edges of connected graph g.

Let G be a finite group, and let $$\textrm{cd}(G)$$ denote the set of degrees irreducible complex characters G. Define then character degree graph $$\Delta (G)$$ as (simple undirected) whose vertices are prime divisors numbers in , two distinct p, q adjacent if only pq divides some number . This paper continues work, started [8], toward classification non-solvable groups possesses cut-vertex, i...

Let G = (V, E) be a connected graph. Let G = (V, E) be a connected graph. An edge set F ⊂ E is said to be a k-restricted edge cut, if G− F is disconnected and every component of G− F has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is the cardinality of a minimum k-restricted edge cut of G. A graph G is λk-connected, if G contains a k-restricted edge cut. A λk...

We describe a new circulation-based method to determine cuts in an undirected graph. A circulation is an oriented labeling of edges with integers so that at each vertex, the sum of the in-labels equals the sum of out-labels. For an integer k, our approach is based on simple algorithms for sampling a circulation (mod k) uniformly at random. We prove that with high probability, certain dependenci...

The Bisection problem asks for a partition of the vertices of a graph into two equally sized sets, while minimizing the cut size. This is the number of edges connecting the two vertex sets. Bisection has been thoroughly studied in the past. However, only few results have been published that consider the parameterized complexity of this problem. We show that Bisection is FPT w.r.t. the minimum c...

A cactus is a connected graph in which any two cycles have at most one vertex in common. We determine the unique graphs with maximum signless Laplacian spectral radius in the class of cacti with given number of cycles (cut edges, respectively) as well as in the class of cacti with perfect matchings and given number of cycles.

We construct a polyhedron that is topologically convex (i.e., has the graph of a convex polyhedron) yet has no vertex unfolding: no matter how we cut along the edges and keep faces attached at vertices to form a connected (hinged) surface, the surface necessarily unfolds with overlap.

A (k; g)-cage is a graph of minimum order among k-regular graphs with girth g. We show that for every cutset S of a (k; g)-cage G, the induced subgraphG[S] has diameter at least bg/2c, with equality only when distance bg/2c occurs for at least two pairs of vertices in G[S]. This structural property is used to prove that every (k; g)-cage with k ≥ 3 is 3-connected. This result supports the conje...

We show that a directed graph extends to a strongly connected digraph on the same vertex set if and only if it has no complete directed cut. We bound the number of edges needed for such an extension, and we apply the characterization to a problem on non-transitive dice.

Hajnal, A. and N. Sauer, Cut-sets in infinite graphs and partial orders. Discrete Mathematics 117 (1993) 113-125. The set S c V(U) is a cut-set of the vertex v of a graph 9 if v is not adjacent to any vertex in S and, for every maximal clique C of Q, ({v} u S) n C # 0. S is a cut-set of the element v of a partial order 9 if S is a cut-set of v in the comparability graph of 8. Given upper bounds...