We introduce a modification of the Harary index where the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.

Two graphs G and H are hypomorphic if there exists a bijection φ : V (G)→ V (H) such that G− v ∼= H − φ(v) for each v ∈ V (G). A graph G is reconstructible if H ∼= G for all H hypomorphic to G. It is well known that not all infinite graphs are reconstructible. However, the Harary-Schwenk-Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we con...

It is shown that the Shields–Harary index of vulnerability of the complete bipartite graph Km,n, with respect to the cost function f (x)= 1− x, 0 x 1, is m, if n m+ 2√m, and 1 n+1 (n+m) 2 4 , ifm n<m+ 2 √ m. It follows that the Shields–Harary number ofKm,n with respect to any concave continuous cost function f on [0, 1] satisfying f (1)=0 ismf (0), if n m+2√m, and between 1 n+1 (n+m) 2 4 f (0) ...

We show that the square of every connected S(K1,4)-free graph satisfying a matching condition has 2-connected spanning subgraph maximum degree at most 3. Furthermore, we characterise trees whose k. This generalises results on S(K1,3)-free graphs Henry and Vogler [7] Harary Schwenk [6], respectively.

Following Harary, the conditional connectivity (edge-connectivity) of a graph with respect to a given graph-theoretic property is the minimum cardinality of a set of vertices (edges), if any, whose deletion disconnects the graph and every remaining component has such a property. We study the case in which all these components are different from a tree whose order is not greater than n. For inst...

There is a conjecture due to Shaoji 3], about uniquely vertex r-colorable graphs which states: \ If G is a uniquely vertex r-colorable graph with order n and size (r ? 1)n ? ? r 2 , then G contains a K r as its subgraph." In this paper for any natural number r we construct a K r-free, uniquely r-colorable graph with (r ? 1)n ? ? r 2 edges. These families of graphs are indeed counter examples to...

Let G be a graph of order n(G), minimum degree (G) and connectivity (G). Chartrand and Harary [Graphs with prescribed connectivities, in: P. Erdös, G. Katona (Eds.), Theory of Graphs, Academic Press, NewYork, 1968, pp. 61–63] gave the following lower bound on the connectivity (G) 2 (G)+ 2− n(G). Topp and Volkmann [Sufficient conditions for equality of connectivity and minimum degree of a graph,...

The Ramsey number r(H,G) is defined as the minimum N such that for any coloring of the edges of the N -vertex complete graph KN in red and blue, it must contain either a red H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K3, G) ≤ 2q + 1 where G has q edges. In other words, any graph on 2q + 1 vertices with independence number at most 2 contains every (iso...

In a paper of Harary and Plantholt, they concluded by noting that they knew of no generalization of the leaf edge exchange (LEE) transition sequence result on spanning trees to other natural families of spanning subgraphs. Now, we give two approaches for such a generalization. We define two kinds of LEE-graphs over the set of all connected spanning k-edge subgraphs of a connected graph G, and s...