the reverse degree distance of a connected graph $g$ is defined in discrete mathematical chemistry as [ r (g)=2(n-1)md-sum_{uin v(g)}d_g(u)d_g(u), ] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $g$, respectively, $d_g(u)$ is the degree of vertex $u$, $d_g(u)$ is the sum of distance between vertex $u$ and all other vertices of $g$, and $v(g)$ is the ...

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

hansen et‎. ‎al.‎, ‎using the autographix software ‎package‎, ‎conjectured that the szeged index $sz(g)$ and the‎ ‎wiener index $w(g)$ of a connected bipartite graph $g$ with $n geq ‎4$ vertices and $m geq n$ edges‎, ‎obeys the relation‎ ‎$sz(g)-w(g) geq 4n-8$‎. ‎moreover‎, ‎this bound would be the best possible‎. ‎this paper offers a proof to this conjecture‎.

We determine, asymptotically in n, the distribution of the weight of a minimum-weight k-clique (or any strictly balanced graph H) in a randomly edge-weighted complete graph Kn.

Given a graph G and a set X ⊆ V (G), the relative Wiener index of X in G is defined as WX(G) = ∑ {u,v}∈(X2 ) dG(u, v) . The graphs G (of even order) in which for every partition V (G) = V1+V2 of the vertex set V (G) such that |V1| = |V2| we have WV1(G) = WV2(G) are called equal opportunity graphs. In this note we prove that a graph G of even order is an equal opportunity graph if and only if it...

Recently, Chan et al. have proposed a new cycle embedding property called balanced pancyclicity [Discrete Applied Mathematics 155(15) (2007) 1971-1978]. For a graph G(V, E) and any two nodes x and y of V, a cycle R contains x and y can be divided into two paths, Pt1 and Pt2, joining x and y such that len(Pt1)len(Pt2), where len() denote the length of the path . A balanced cycle of an even (r...

Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n/v copies of H whose vertex sets partition V (G). In this paper we consider the threshold th H (n) of the property that an Erd˝ os-Rényi random graph (on n points) contains an H-factor. Our results determine th H (n) for all strictly balanced H. The method here extends with ...

the emph{harary index} $h(g)$ of a connected graph $g$ is defined as $h(g)=sum_{u,vin v(g)}frac{1}{d_g(u,v)}$ where $d_g(u,v)$ is the distance between vertices $u$ and $v$ of $g$. the steiner distance in a graph, introduced by chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. for a connected graph $g$ of order at least $2$ and $ssubseteq v(g)$, th...

Definition. Let Kn denote the complete graph of n vertices. The complete multigraph λKn is the complete graph Kn in which every edge is taken λ times. The t-foil (or the t-windmill) is a graph of t edge-disjoint K3’s with a common vertex and the common vertex is called the center of the t-foil. In particular, the 2-foil, the 3-foil, and the 4-foil are called the bowtie, the trefoil, and the qua...