We have used a scanning laser technique to measure in vitro changes in back vertex distance of chick and pigeon lenses. Enucleated eyes were dissected, leaving the lens naturally suspended by the ciliary body and intraocular muscles. Ray tracing techniques were used to measure the resting back vertex distance of the lenses by passing a laser beam through the lens and scanning it across the pupi...

As highly discriminant distance-based topological indices, the Balaban index and the sum-Balaban index of a graph $G$ are defined as $J(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=sumlimits_{vin V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the n...

The guided tree edit distance problem is to find a minimum cost series of edit operations that transforms two input forests F and G into isomorphic forests F ′ and G′ such that a third input forest H is included in F ′ (and G′). The edit operations are relabeling a vertex and deleting a vertex. We show efficient algorithms for this problem that are faster than the previous algorithm for this pr...

A graph is shown to be distance-hereditary if and only if no induced subgraph of order five or more has a unique hamiltonian cycle; this is also equivalent to every induced subgraph of order five or more having an even number of hamiltonian cycles. Restricting the induced subgraphs to those of odd order five or more gives two similar characterizations of parity graphs. The close relationship be...

We address the chromatic number of a class of Euclidean distance graphs having vertex set Q, the 3-dimensional rational space. It is shown that if s is any positive integer with no prime factors congruent to 2 (mod 3), and G is the graph with vertex set Q where any two vertices are adjacent if and only if they are distance √ 2s apart, then G has chromatic number 4. Along the way, we obtain a fe...

In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. 2 ) ( = G β ) is characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever 2 ) (...

Let G = (V,E) be a graph of order n. Let f : V → {1, 2, . . . , n} be a bijection. For any vertex v ∈ V , the neighbor sum ∑u∈N(v) f(u) is called the weight of the vertex v and is denoted by w(v). If w(v) = k, (a constant) for all v ∈ V , then f is called a distance magic labeling with magic constant k. If the set of vertex weights forms an arithmetic progression {a, a+ d, a+2d, . . . , a+ (n− ...

Due to exponential growth of complex data, graph structure has become increasingly important to model various entities and their interactions, with many interesting applications including, bioinformatics, social network analysis, etc. Depending on the complexity of the data, the underlying graph model can be a simple directed/undirected and/or weighted/un-weighted graph to a complex graph (aka ...

A set D of vertices in a connected graph G is called a k-dominating set if every vertex in G − D is within distance k from some vertex of D. The k-domination number of G, γk(G), is the minimum cardinality over all k-dominating sets of G. A graph G is k-distance domination-critical if γk(G − x) < γk(G) for any vertex x in G. This work considers properties of k-distance domination-critical graphs...

A vertex v of a graph G is a boundary vertex if there exists a vertex u such that the distance in G from u to v is at least the distance from u to any neighbour of v. We give a full description of all graphs that have exactly four boundary vertices, which answers a question of Hasegawa and Saito. To this end, we introduce the concept of frame of a graph. It allows us to construct, for every pos...