<p>A graph denoted by is a pair of where non-empty set vertices in G, and E edges G. In theory, there are various types graphs including star graphs, cycle graph, wheel graph. Graph operations on two or more can produce new graphs. Amalgamation one the Suppose connected amalgamating 1 2 joining obtained joins vertex into , common resulting from while edge combining g, g side (

Given a graph with costs on the edges, the power of a node is the maximum cost of an edge leaving it, and the power of the graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several network design problems under the power minimization criteria. Given a graph G = (V, E) with costs on the edges and requirements r(v) for each v ∈ V , the Min-Po...

For two vertices u and v in a graph G = (V, E), the detour distance D(u, v) is the length of a longest u–v path in G. A u–v path of length D(u, v) is called a u–v detour. A set S ⊆ V is called an edge detour set if every edge in G lies on a detour joining a pair of vertices of S. The edge detour number dn1(G) of G is the minimum order of its edge detour sets and any edge detour set of order dn1...

For this paper, we will define a (non-oriented) graph Γ to be a pair Γ = (V,E), where V = vert(Γ) is a set of vertices, and E = edge(Γ) ⊆ V × V/S2 is a set of unordered pairs, known as edges between them. Two vertices, v, v′ ∈ V are considered adjacent if (v, v′) ∈ E, if there is an edge between them. An oriented graph has edge set E = edge(Γ) ⊆ V × V , ordered pairs. For an edge v = (v1, v2) i...

Let G=(V(G),E(G)) be a connected simple undirected graph with non empty vertex set V(G) and edge set E(G). For a positive integer k, by an edge irregular total k-labeling we mean a function f : V(G)UE(G) --> {1,2,...,k} such that for each two edges ab and cd, it follows that f(a)+f(ab)+f(b) is different from f(c)+f(cd)+f(d), i.e. every two edges have distinct weights. The minimum k for which G ...

A concept related to the spectrum of a graph is that of energy. The energy E(G) of a graph G is equal to the sum of the absolute values of the eigenvalues of the adjacency matrix of G . The Laplacian energy of a graph G is equal to the sum of distances of the Laplacian eigenvalues of G and the average degree d(G) of G. In this paper we introduce the concept of Laplacian energy of fuzzy graphs. ...

Let G = (V + s,E) be a graph and let S = (d1, ..., dp) be a set of positive integers with ∑ dj = d(s). An S-detachment splits s into a set of p independent vertices s1, ..., sp with d(sj) = dj , 1 ≤ j ≤ p. Given a requirement function r(u, v) on pairs of vertices of V , an S-detachment is called r-admissible if the detached graph G satisfies λG′(x, y) ≥ r(x, y) for every pair x, y ∈ V . Here λH...

Graph Connectivity and Network Coding LEUNG, Kai Man Master of Philosophy Department of Computer Science and Engineering The Chinese University of Hong Kong 2011 In this thesis we present a new algebraic formulation to compute edge connectivities in a directed graph, using the ideas developed in network coding. This reduces the problem of computing edge connectivities to solving systems of line...

For every h ∈ N, a graph G with the vertex set V (G) and the edge set E(G) is said to be h-magic if there exists a labeling l : E(G) → Zh\{0} such that the induced vertex labeling s : V (G) → Zh, defined by s(v) = ∑ uv∈E(G) l(uv) is a constant map. When this constant is zero, we say that G admits a zero-sum h-magic labeling. The null set of a graph G, denoted by N(G), is the set of all natural ...

ABSTRACTIntroduction: Cancer is caused by genetic abnormalities, such as mutation of ontogenesis or tumor suppressor genes which alter downstream signaling pathways and protein-protein interactions. Comparison of protein interactions in cancerous and normal cells can be of help in mechanisms of disease diagnoses and treatments. Methods: We constructed protein interaction networks of cancerous a...