Graph Equations for Line Graphs, Jump Graphs, Middle Graphs, Splitting Graphs And Line Splitting Graphs

For a graph G, let G, L(G), J(G) S(G), L,(G) and M(G) denote Complement, Line graph, Jump graph, Splitting graph, Line splitting graph and Middle graph respectively. In this paper, we solve the graph equations L(G) =S(H), M(G) = S(H), L(G) = LS(H), M(G) =LS(H), J(G) = S(H), M(G) = S(H), J(G) = LS(H) and M(G) = LS(G). The equality symbol '=' stands for on isomorphism between two graphs

Equitable total domination in graphs

A subset ‫ܦ‬ of a vertex set ܸሺ‫ܩ‬ሻ of a graph ‫ܩ‬ ൌ ሺܸ, ‫ܧ‬ሻ is called an equitable dominating set if for every vertex ‫ݒ‬ ‫א‬ ܸ െ ‫ܦ‬ there exists a vertex ‫ݑ‬ ‫א‬ ‫ܦ‬ such that ‫ݒݑ‬ ‫א‬ ‫ܧ‬ሺ‫ܩ‬ሻ and |݀݁݃ሺ‫ݑ‬ሻ െ ݀݁݃ሺ‫ݒ‬ሻ| 1, where ݀݁݃ሺ‫ݑ‬ሻ and ݀݁݃ሺ‫ݒ‬ሻ are denoted as the degree of a vertex ‫ݑ‬ and ‫ݒ‬ respectively. The equitable domination number of a graph ߛ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable dominating set of ‫.ܩ‬ An equitable dominating set ‫ܦ‬ is said to be an equitable total dominating set if the induced subgraph ‫ۄܦۃ‬ has no isolated vertices. The equitable total domination number ߛ ௧ ሺ‫ܩ‬ሻ of ‫ܩ‬ is the minimum cardinality of an equitable total dominating set of ‫.ܩ‬ In this paper, we initiate a study on new domination parameter equitable total domination number of a graph, characterization is given for equitable total dominating set is minimal and also discussed Northaus-Gaddum type results

k-trees, k-ctrees and line splitting graphs

Let G = (V,E) be a graph. For each edge ei of G, a new vertexe?i is taken and the resulting set of vertices is denoted by E1(G). Theline splitting graph Ls(G) of a graph G is defined as the graph havingvertex set E(G)SE1(G) with two vertices adjacent if they correspondto adjacent edges of G or one corresponds to an element e?i of E1(G)and the other to an element ej of E(G) where ej is in N(ei). In thispaper we characterize graphs whose line splitting graphs are k ? treesand k ? ctrees

Characterizations of planar plick graphs

In this paper we present characterizations of graphs whose plick graphs are planar, outerplanar and minimally nonouterplanar

Multiplicative Zagreb indices and coindices of some derived graphs

In this note, we obtain the expressions for multiplicative Zagreb indices and coindices of derived graphs such as a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph and paraline graph

Integrity of Wheel Related Graphs

If a network modeled by a graph, then there are various graphtheoretical parameters used to express the vulnerability of communicationnetworks. One of them is the concept of integrity. In this paper, we determineexact values for the integrity of wheel related graphs

Connected cototal domination number of a graph

A dominating set $D subseteq V$ of a graph $G = (V,E)$ is said to be a connected cototal dominating set if $langle D rangle$ is connected and $langle V-D rangle neq phi$, contains no isolated vertices. A connected cototal dominating set is said to be minimal if no proper subset of $D$ is connected cototal dominating set. The connected cototal domination number $gamma_{ccl}(G)$ of $G$ is the minimum cardinality of a minimal connected cototal dominating set of $G$. In this paper, we begin an investigation of connected cototal domination number and obtain some interesting results

Multiplicative Zagreb Indices of Generalized Transformation Graphs

The rst, second and modied rst multiplicative Zagreb indicesof a graph G are dened, respectively, as Π1(G) =Πu2V (G)dG(u)2, Π2(G) =Πuv2E(G) dG(u)dG(v)and Π 1(G) =Πuv2E(G)[dG(u) + dG(v)] where dG(w) is the degree of vertex w in G. In the present study, we obtain the expressions for Π1,Π 2 and Π 1 of generalized transformation graphs Gab