Recently, Hua et al. defined a new topological index based on degrees and inverse of distances between all pairs of vertices. They named this new graph invariant as reciprocal degree distance as 1 { , } ( ) ( ( ) ( ))[ ( , )] RDD(G) = u v V G d u  d v d u v , where the d(u,v) denotes the distance between vertices u and v. In this paper, we compute this topological index for Grassmann graphs.

A graph H is an edge reconstruction of the graph G if there is a bijection p from E(G) to E(H) such that for each edge e in E(G), G\e is isomorphic to EI\fi(e). (Here G\e denotes the graph obtained from G by deleting the edge e.) We call G edge reconstructible if any edge reconstruction of G is isomorphic to G. The well-known edge-reconstruction conjecture, due to Harary [4], asserts that any g...

There are a number of positional games known on the infinite chessboard. One of the most studied is the 5-in-a-row, whose rules are almost identical to the ancient Japanese Go-Moku. Along this line Harary asked if a player can achieve a translated copy of a given polymino P when the two players alternately take the squares of the board. Here we pose his question for general subsets of the board...

In this work we deal with questions on balance and weak balance in random graphs. The theory of balance goes back to Heider [12] who asserted that a social system is balanced if there is no tension and that unbalanced social structures exhibit a tension resulting in a tendency to change in the direction of balance. Since this first work of Heider, the notion of balance has been extensively stud...

In 1967 Harary, Hedetniemi, and Prins showed that every graph G admits a complete t-coloring for t with χ(G)≤t≤ψ(G), where χ(G) denotes the chromatic number of ψ(G) achromatic which is maximum r r-coloring. Recently, Edwards Rza̧żewski (2020) this result fails hypergraphs by proving integer k k≥9, there exists k-uniform hypergraph H χ(H)-coloring ψ(H)-coloring, but no some χ(H)<t<ψ(H). They also...

We show that the geodetic game, introduced by Fraenkel and Harary, is decidable in polynomial time on AT-free graphs.

For any two vertices u and v in a graph G (digraph D, respectively), a u − v geodesic is a shortest path between u and v (from u to v, respectively). Let I(u, v) denote the set of all vertices lying on a u− v geodesic. For a vertex subset S, let I(S) denote the union of all I(u, v) for u, v ∈ S. The geodetic number g(G) (g(D), respectively) of a graph G (digraph D, respectively) is the minimum ...

The eccentricity of a vertex is the maximum distance from it to another vertex, and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we introduce average edge and average vertex-edge mean eccentricities of a graph. Moreover, relations among these eccentricities for trees are provided as well as formulas for line graphs and cartesian product of graphs. In t...

We generalize the Harary-Sachs theorem to k-uniform hypergraphs: codegree-d coefficient of characteristic polynomial a uniform hypergraph H can be expressed as weighted sum subgraph counts over certain multi-hypergraphs with d edges. include detailed description aforementioned and formula for their corresponding weights.