A labeling of edges and vertices a simple graph \(G(V,E)\) by mapping \(\Lambda :V\left( G \right) \cup E\left( \to \left\{ { 1,2,3, \ldots ,\Psi } \right\}\) provided that any two pair have distinct weights is called an edge irregular total \(\Psi\)-labeling. If \(\Psi\) minimum \(G\) admits -labelling, then the irregularity strength (TEIS) denoted \(\mathrm{tes}\left(G\right).\) In this paper...

Let G be a planar graph of maximum degree ∆ and girth g, and there is an integer t(> g) such that G has no cycles of length from g+1 to t. Then the total chromatic number of G is ∆+1 if (∆,g, t) ∈ {(5,4,6),(4,4,17)}; or ∆ = 3 and (g, t) ∈ {(5,13),(6,11),(7,11), (8,10),(9,10)}, where each vertex is incident with at most one g-cycle. 2010 Mathematics Subject Classification: 05C15

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.

An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that any pair of adjacent vertices are incident to distinct sets of colors. The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by χ′′ a(G). Let mad(G) and ∆(G) denote the maximum average degree and the maximum degree of a graph G, respectivel...