Let G = ( V , E ) be a simple, connected graph. One is often interested in short path between two vertices u v ∈ . We propose spectral algorithm: construct the function ϕ : → R ≥ 0 arg ⁡ min f ≢ ∑ w 1 2 − can also understood as smallest eigenvector of Laplacian Matrix L D A after -th row and column have been removed. start point from to at each step, we move neighbor for which smallest. This al...

In this paper we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the socalled generalized tree shift which seems to be a powerful tool to attack the problems of the above mentioned kind. We show that the generalized tree shift increases the l...

The simulated annealing algorithm has been introduced into the broadcast routing of wireless sensor network so as to reduce the length and energy of broadcast path. However, its new solution is generated by randomly exchanging the location of two sensor nodes of broadcast path. Though the simulated annealing algorithm is better than 2-opt algorithm, its method of generating new solution has sti...

Nakano et al. in [20] presented a timeand work-optimal algorithm for finding the smallest number of vertex-disjoint paths that cover the vertices of a cograph and left open the problem of applying their technique into other classes of graphs. Motivated by this issue we generalize their technique and apply it to the class of P4-sparse graphs, which forms a proper superclass of cographs. We show ...

Nowadays most data networks use shortest path protocols such as OSPF or IS-IS to route traffic. Given administrative routing lengths for the links of a network, all data packets are sent along shortest paths with respect to these lengths from their source to their destination. One of the most fundamental problems in planning shortest path networks is to decide whether a given set of routing pat...

For a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is non-zero whenever i / =j and {i, j} is an edge in G. The path cover number of G is the minimum number of vertexdisjoint paths occurring as induced subgraphs of G that cover all the vertices of G. For trees, the relationship between minimum ...