Let G be a simple graph. A total dominator coloring of G, is a proper coloring of the vertices of G in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number χ t d (G) of G, is the minimum number of colors among all total dominator coloring of G. The neighbourhood corona of two graphs G 1 and G 2 is denoted by G 1 ⋆ G 2 and is the graph obtained by taking one copy of G 1 and |V (G 1 )| copies of G 2 , and joining the neighbours of the ith vertex of G 1 to every vertex in the ith copy of G 2 . In this paper, we study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of r-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of G is the minimum number of vertices (edges) of G whose removal changes the TDC-number of G. We study the stability and bondage number of certatin graphs.

Let G be a finite connected graph of order n. The Gutman index Gut(G) of G is defined as ∑ {x,y}⊆V (G) deg(x)deg(y)d(x, y), where deg(x) is the degree of vertex x in G and d(x, y) is the distance between vertices x and y in G. A cactus graph is a connected graph in which no edge lies in more than one cycle. In this paper we compute the exact value of Gutman index of some cactus chains.

, where d u is the degree of vertex u in G. In this paper, we study this index for certain graphs and we examine the effects on SO(G) when G is modified by operations on vertex and edge of G. Also we present bounds for the Sombor index of join and corona product of two graphs.

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Let G be a simple graph of order n. The energy E(G) of the graph G is the sum of the absolute values of the eigenvalues of G. The Randić matrix of G, denoted by R(G), is defined as the n × n matrix whoseif v i and v j are adjacent and 0 for another cases. The Randić energy RE of G is the sum of absolute values of the eigenvalues of R(G). In this paper we compute the energy and Randić energy for certain graphs. Also we propose a conjecture on Randić energy.Mathematics Subject Classification: 15A18.