A set D of vertices in a graph G is a dominating set if every vertex in V −D is adjacent to some vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A dominating set D of a graph G is total dominating set if the induced subgraph 〈D〉 has no isolated vertices. In this paper, we introduce the total co-independent domination in graphs, exact value for some s...

In this paper we survey the behavior of various graph operations on the graph parameters clique-width and NLC-width. We give upper and lower bounds for the clique-width and NLC-width of the modi ed graphs in terms of the clique-width and NLC-width of the involved graphs. Therefor we consider the binary graph operations join, co-join, sum, di erence, products, corona, substitution, and 1-sum, an...

In this paper, at first we introduce a new index with the name Co-PI index and obtain some properties related this new index. Then we compute this new index for TUC4C8(R) nanotubes.

the zagreb indices are the oldest graph invariants used in mathematical chemistry to predictthe chemical phenomena. in this paper we define the multiple versions of zagreb indicesbased on degrees of vertices in a given graph and then we compute the first and secondextremal graphs for them.

let $g$ be a connected graph with vertex set $v(g)$‎. ‎the‎ ‎degree resistance distance of $g$ is defined as $d_r(g) = sum_{{u‎,‎v} subseteq v(g)} [d(u)+d(v)] r(u,v)$‎, ‎where $d(u)$ is the degree‎ ‎of vertex $u$‎, ‎and $r(u,v)$ denotes the resistance distance between‎ ‎$u$ and $v$‎. ‎in this paper‎, ‎we characterize $n$-vertex unicyclic‎ ‎graphs having minimum and second minimum degree resista...

We give a simple proof based on symmetries that there are no geodesics from vertex to itself in the cube, tetrahedron, octahedron, and icosahedron.

Structurally cohesive subgroups are a powerful and mathematically rigorous way to characterize network robustness. Their strength lies in the ability to detect strong connections among vertices that not only have no neighbors in common, but that may be distantly separated in the graph. Unfortunately, identifying cohesive subgroups is a computationally intensive problem, which has limited empiri...

We propose to mine the topology of a large attributed graph by finding regularities among vertex descriptors. Such descriptors are of two types: (1) the vertex attributes that convey the information of the vertices themselves and (2) some topological properties used to describe the connectivity of the vertices. These descriptors are mostly of numerical or ordinal types and their similarity can ...

‎Recently‎, ‎some techniques such as adding whiskers and attaching graphs to vertices of a given graph‎, ‎have been proposed for constructing a new vertex decomposable graph‎. ‎In this paper‎, ‎we present a new method for constructing vertex decomposable graphs‎. ‎Then we use this construction to generalize the result due to Cook and Nagel‎.