The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlike-threshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge c...

A graph is said to be determined by its signless Laplacian spectrum if there is no other non-isomorphic graph with the same spectrum. In this paper, it is shown that each starlike tree with maximum degree 4 is determined by its signless Laplacian spectrum.

In the graph searching problem initially a graph with all edges contaminated is pre sented We would like to obtain a state of the graph in which all edges are simultaneously clear by a sequence of moves using searchers The objective is to achieve the desired state by using the least number of searchers Two variations of the graph searching problem are considered in this paper One is the edge se...

Let G(V, E) be a graph with n-vertex-set V and m-edge-set E. Meanwhile, each vertex v is associated with a nonnegative weight W(v). Given a subset F of V, this paper studies the problem of finding a perfect dominating set D such that δ(D) = ∑ is minimized under the restriction that D ⊆ (V F). The vertices in F are called forbidden vertices. We first define a new class of graphs, called l ∈D v D...

The reverse Wiener index of a connected graph G is defined as Λ(G) = 1 2 n(n− 1)d−W (G), where n is the number of vertices, d is the diameter, and W (G) is the Wiener index (the sum of distances between all unordered pairs of vertices) of G. We determine the n-vertex non-starlike trees with the first four largest reverse Wiener indices for n > 8, and the nvertex non-starlike non-caterpillar tre...

A starlike tree (or a quasistar) is a subdivision of a star tree. A family of hypercube-like interconnection networks called restricted HLgraphs includes many interconnection networks proposed in the literature such as twisted cubes, crossed cubes, multiply twisted cubes, Möbius cubes, Mcubes, and generalized twisted cubes. We show in this paper that every starlike tree of degree at most m with...

The transmission of a vertex v graph G is the sum distances from to all other vertices in . A irregular if its have pairwise different transmissions. starlike tree T ( k 1 , … t ) obtained by attaching an isolated pendant paths lengths respectively. It proved that + ? 2 odd order, then it irregular. ? 3 and only ? { r : } Additional infinite families among trees bi-starlike are determined. Tran...

The INDUCED MINOR problem is to test whether a graph G contains a graph H as an induced minor, i.e., if G can be modified into H by a sequence of vertex deletions and edge contractions. When H is fixed, i.e., not part of the input, this problem is denoted H -INDUCED MINOR. We provide polynomialtime algorithms for this problem in the case that the fixed target graph has a starlike structure. In ...

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means subordination, analytic parts which are reciprocal starlike (or convex) functions. Further, combining graph function, discuss bound Bloch constant and norm pre-Schwarzian derivative for classes.

A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS) if every with same isomorphic G. In some recent papers it proved that friendship graphs and starlike trees are DLS. If a tree joined merging their vertices degree greater than two, then resulting called path-friendship graph. this paper, graphs, natural generalization trees, also Consequently, using these results w...