Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f : V (G) → {0, 1, 2, · · · , k} with the following satisfied for all vertices u and v: |f(u)−f(v)| ≥ diam(G)−dG(u, v)+1, where dG(u, v) is the distancee between u and v. We prove a lower bound for the radio number of trees, and characterize the...

We prove a necessary and sufficient condition under which a connected graph has a connected P3-path graph. Moreover, an analogous condition for connectivity of the Pk-path graph of a connected graph which does not contain a cycle of length smaller than k+1 is derived.

where ni = nβ 2 i (i = 1, 2); β1 and β2 denote the main angles of μ1 and μ2, respectively. Further, let G be any connected or disconnected graph (not necessarily with two main eigenvalues). Let S be any subset of the vertex set V (G) and let GS be the graph obtained from the graph G by adding a new vertex x which is adjacent exactly to the vertices from S. If σ(GS1) = σ(GS2) then we prove that ...

A (δ, g)-cage is a regular graph of degree δ and girth g with the least possible number of vertices. It was proved by Fu, Huang and Rodger that every (3, g)-cage is 3-connected. Moreover, the same authors conjectured that all (δ, g)-cages are δ-connected for every δ ≥ 3. As a first step towards the proof of this conjecture, Jiang and Mubayi showed that every (δ, g)-cage with δ ≥ 3 is 3-connecte...

In 1956, W.T. Tutte proved that a 4-connected planar graph is hamiltonian. Moreover, in 1997, D.P. Sanders extended this to the result that a 4-connected planar graph contains a hamiltonian cycle through any two of its edges. We prove that a planar graph G has a cycle containing a given subset X of its vertex set and any two prescribed edges of the subgraph of G induced by X if |X| ≥ 3 and if X...

if $g$ is a connected graph with vertex set $v$, then the eccentric connectivity index of $g$, $xi^c(g)$, is defined as $sum_{vin v(g)}deg(v)ecc(v)$ where $deg(v)$ is the degree of a vertex $v$ and $ecc(v)$ is its eccentricity. in this paper we show some convergence in probability and an asymptotic normality based on this index in random bucket recursive trees.

In the fully dynamic graph connectivity problem we are modifying a graph by inserting and deleting edges and answering queries if two vertices are connected by a path. For the offline case we are given all of the queries and modifications beforehand, while for the online we should respond without any knowledge for the future operations. In this paper we are going to compare implementation of th...

In this paper, we present some new lower and upper bounds for the modified Randic index in terms of maximum, minimum degree, girth, algebraic connectivity, diameter and average distance. Also we obtained relations between this index with Harmonic and Atom-bond connectivity indices. Finally, as an application we computed this index for some classes of nano-structures and linear chains.

Let M and N be internally 4-connected binary matroids such that M has a proper N -minor, and |E(N)| ≥ 7. As part of our project to develop a splitter theorem for internally 4-connected binary matroids, we prove the following result: if M\e has no N -minor whenever e is in a triangle of M , and M/e has no N -minor whenever e is in a triad of M , then M has a minor, M ′, such that M ′ is internal...

Let G(n; m) be a connected graph without loops and multiple edges which has n vertices and m edges. We ÿnd the graphs on which the zeroth-order connectivity index, equal to the sum of degrees of vertices of G(n; m) raised to the power − 1 2 , attains maximum.