The Randić index R(G) of a graph G is defined as the sum of 1 √dudv over all edges uv of G, where du and dv are the degrees of vertices u and v, respectively. Let D(G) be the diameter of Gwhen G is connected. Aouchiche et al. (2007) [1] conjectured that among all connected graphs G on n vertices the path Pn achieves the minimum values for both R(G)/D(G) and R(G) − D(G). We prove this conjecture...

Given a connected graph G = (V,E), two players take turns occupying vertices v ∈ V by putting black and white tokens so that the current vertex sets B,W ⊆ V are disjoint, B ∩W = ∅, and the corresponding induced subgraphs G[B] and G[W ] are connected any time. A player must pass whenever (s)he has no legal move. (Obviously, after this, the opponent will take all remaining vertices, since G is as...

The antipodal graph A(G) of a graph G is defined as the graph on the same vertex set as G with two vertices being adjacent in A(G) if the distance between them in G is the diameter of G. (If G is disconnected then we define &am(G) = co.) Aravamudhan and Rajendran [l, 21 gave the following characterization of antipo-da1 graphs. Theorem 1. A graph G is an antipodal graph ifand only ifit is the an...

The reconstruction number rn(G) of graph G is the minimum number of vertex-deleted subgraphs of G required in order to identify G up to isomporphism. Myrvold and Molina have shown that if G is disconnected and not all components are isomorphic then rn(G) = 3, whereas, if all components are isomorphic and have c vertices each, then rn(G) can be as large as c + 2. In this paper we propose and ini...

The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $chi_{alpha}$ values from a certain collection of polyomino ...

Let be a simple and connected graph. A k-vertex separator [k-edge separator] is a subset of vertices [edges] whose deletion separates the vertex [edge] set of into two parts of equal cardinality, that are at distance greater than k in . Here we investigate the relation between the cardinality of these cutsets and the laplacian spectrum of . As a consequence of the study, we obtain the well-know...

Let G be a graph and u, v be two distinct vertices of G. A u−v path P is called nonseparating if G − V (P ) is connected. The purpose of this paper is to study the number of nonseparating u− v path for two arbitrary vertices u and v of a given graph. For a positive integer k, we will show that there is a minimum integer α(k) so that if G is an α(k)-connected graph and u and v are two arbitrary ...

Let (e,, e2, e3, e4) be the standard basis of R4. The 24-ceI& is a 4-dimensional regular polytope whose vertices are the 24 vectors *q =te, (i # i) of R4, two vertices being adjacent iff the angle between the corresponding vectors is 60’. It is well known [2] and easy to check that the l-skeleton (i.e. the graph consisting of the vertices and edges) of this polytope has the following property: ...

Denote by an l-component a connected graph with l edges more than vertices. We prove that the expected number of creations of (l + 1)-component, by means of adding a new edge to an l-component in a randomly growing graph with n vertices, tends to 1 as l, n tends to ∞ but with l = o(n1/4). We also show, under the same conditions on l and n, that the expected number of vertices that ever belong t...

By a 2-packing in a graph wemean a subset of its vertex set, in which all the vertices are in distance at least 3 from each other. The question about the maximum number of 2-packings in a graph is strongly related to the problem of L(2, 1)-labeling of graphs. In this paperwe findnewasymptotic upper and lower bounds on themaximumnumber of 2-packings in a connected graph on n vertices. The bounds...