The edge Szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph G, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. In this paper, the edge Szeged index of one-pentagonal carbon nanocone CNC5[n] is computed for the first ...

A set $Wsubseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,vin V(G)$ there exists $win W$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $dim(G)$. In this paper, it is proved that in a connected graph $...

We prove Ramsey-type results for intersection graphs of geometric objects the plane. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n ≥ 2 convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a con...

Trees are a fundamental structure in algorithmics. In this paper we study the transformation of an arbitrary binary tree S with n vertices into a completely balanced tree T via rotations, a widely studied elementary tree operation. Combining concepts on rotation distance and data structures we give a basic algorithm that performs the transformation in Θ(n) time and Θ(1) space, making at most 2n...

The concept of strongly balanced graph is introduced. It is shown that there exists a strongly balanced graph with u vertices and e edges if and only if I s u 1 s e s (2"). This result is applied to a classic question of Erdos and Renyi: What is the probability that a random graph on n vertices contains a given graph? A rooted version of this problem is also solved.

We study the problem of partitioning a small sample of n individuals from a mixture of k product distributions over a Boolean cube {0, 1}K according to their distributions. Each distribution is described by a vector of allele frequencies in RK . Given two distributions, we use γ to denote the average l2 distance in frequencies across K dimensions, which measures the statistical divergence betwe...

A signed graph is a graph whose edges are given ±1 weights. In such a graph, the sign of a cycle is the product of the signs of its edges. A signed graph is called balanced if its adjacency matrix is similar to the adjacency matrix of an unsigned graph via conjugation by a diagonal ±1 matrix. For a signed graph Σ on n vertices, its exterior kth power, where k = 1, . . . , n−1, is a graph ∧k Σ w...

Mantel’s theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let K− 4 = {123, 124, 134}, F6 = {123, 124, 345, 156} and F = {K− 4 , F6}: for n 6= 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tripartite 3-graph S3(n) (for n ...

Matching preclusion is a measure of robustness in the event of edge failure in interconnection networks. The matching preclusion number of a graph G is the minimum number of edges whose deletion leaves the resulting graph without a perfect matching or an almost perfect matching, and the conditional matching preclusion number of G is the minimum number of edges whose deletion leaves the resultin...