Abstract: Let G be a fuzzy graph. Then D Ôèå V is said to be a strong (weak) fuzzy dominating set of G if every vertex v ∈ V -D is strongly (weakly) dominated by some vertex u in D. We denote a strong (weak) fuzzy dominating set by sfd-set (wfd-set). The minimum scalar cardinality of a sfd-set (wfd-set) is called the strong (weak) fuzzy domination number of G and it is denoted by γsf (G)γwf (G). In this paper we introduce the concept of strong (weak) domination in fuzzy graphs and obtain some interesting results for this new parameter in fuzzy graphs.
Abstract: Let G be a graph of order n. The second stage adjacency matrix of G is the symmetric n × n matrix for which the ijth entry is 1 if the vertices vi and vj are of distance two; otherwise 0. The sum of the absolute values of this second stage adjacency matrix is called the second stage energy of G. In this paper we investigate a few properties and determine some upper bounds for the largest eigenvalue.
Abstract: The Detour matrix (DD) of a graph has for its ( i , j)
entry the length of the longest path between vertices i and j. The
DD-eigenvalues of a connected graph G are the eigenvalues for its
detour matrix, and they form the DD-spectrum of G. The DD-energy
EDD of the graph G is the sum of the absolute values of its DDeigenvalues.
Two connected graphs are said to be DD- equienergetic
if they have equal DD-energies. In this paper, the DD- spectra of a
variety of graphs and their DD-energies are calculated.
Abstract: In this paper, we present a framework to determine Haar solutions of Bratu-type equations that are widely applicable in fuel ignition of the combustion theory and heat transfer. The method is proposed by applying Haar series for the highest derivatives and integrate the series. Several examples are given to confirm the efficiency and the accuracy of the proposed algorithm. The results show that the proposed way is quite reasonable when compared to exact solution.