The spectral closeness of a graph G is defined as the radius matrix , whose ( u v )-entry for vertex and 2 − d u,v ) if ≠ 0 otherwise, where distance between in . residual nontrivial minimum subgraphs with one deleted. We propose local grafting operations that decrease or increase determine those graphs uniquely minimize and/or maximize some families graphs. also discuss extremal properties clo...

The most famous one for finding the shortest path is the Dijkstra, but it has some limitations. For example, when there are more than one shortest path between the source node and one special node, Dijkstra could find only one of them. Besides, the algorithm is quite complex. This article introduce a conception of coding graph, abstracting the problem of shortest paths and critical paths into t...

The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values for the vertex-deleted subgraphs of the same graph. This note discusses the polynomial reconstructibility of the matching polynomial. We collect previous resul...

A graph G is said to be n-factor-critical if G − S has a 1-factor for any S ⊂V (G) with |S| = n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with 3(G)¿ 2 (p−n−1), then G is hamiltonian with some exceptions. To extend this theorem, we de6ne a (k; n)-factor-critical graph to be a graph G such that G − S has a k-factor for any S ⊂V (G) with |S| = n. We con...

We consider the following graph modification problem. Let the input consist of a graph G = (V,E), a weight function w : V ∪E → N, a cost function c : V ∪E → N and a degree function δ : V → N0, together with three integers kv, ke and C. The question is whether we can delete a set of vertices of total weight at most kv and a set of edges of total weight at most ke so that the total cost of the de...

Two preferential attachment type graph models which allow for dynamic addition/deletion of edges/vertices are considered. The focus of this paper is on the limiting expected degree of a fixed vertex. For both models a phase transition is seen to occur, i.e. if the probability with which edges are deleted is below a model specific threshold value, the limiting expected degree is infinite, but if...

In 1941 Kelly and Ulam proposed the Graph Reconstruction Conjecture, and it has remained an open problem to this day[3]. The Graph Reconstruction Conjecture states simply that any simple finite undirected graph on 3 or more vertices can be uniquely identified (up to isomorphism) by the multiset of its unlabeled 1-vertex-deleted subgraphs. There are no known counter-examples to the Conjecture, a...

In the classical vertex graph reconstruction problem a vertex is deleted in every possible way from a graph G, and then it can be asked how many (both minimum and maximum values) of these subgraphs are required to uniquely reconstruct G (up to isomorphism). In the graph k-vertex reconstruction problem k vertices are deleted, and then the same questions can be asked. Data currently exists descri...

For a family of connected graphs A , spanning subgraph H graph G is called an -factor if each component isomorphic to some in . has perfect 2-matching such that either edge or cycle, i.e. { P 2 C i | ≥ 3}-factor said be covered if, for every e ∈ E ( ), there M belongs deleted − possesses 2-matching. In this paper, we first obtain respective new characterizations bipartite and non-bipartite by p...