A total dominator coloring of \(G\) without isolated vertex is a proper coloring of the vertices of \(G\) in which each vertex of \(G\) is adjacent to every vertex of some color class. The total dominator chromatic number \(\chi^t_d(G)\) of \(G\) is the minimum number of colors among all total dominator coloring of \(G\). In this paper, we will give the polynomial time algorithms to computing the total dominator coloring number for \(P_4\)-reducible and \(P_4\)-tidy graphs.

An \(H\)-(a,d)-antimagic labeling in a \(H\)-decomposable graph \(G\) is a bijection \(f: V(G)\cup E(G)\rightarrow {\{1,2,…,p+q\}}\) such that \(\sum f(H_1),\sum f(H_2),\cdots, \sum f(H_h)\) forms an arithmetic progression with difference \(d\) and first element \(a\). \(f\) is said to be \(H\)-\(V\)-super-\((a,d)\)-antimagic if \(f(V(G))={\{1,2,…,p\}}\). Suppose that \(V(G)=U(G) \cup W(G)\) with \(|U(G)|=m\) and \(|W(G)|=n\). Then \(f\) is said to be \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic labeling if \(f(U(G))={\{1,2,…,m\}}\) and \(f(W(G))={\{m+1,m+2,…,(m+n=p)\}}\). A graph that admits a \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic labeling is called a \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic decomposable graph. In this paper, we prove that complete bipartite graphs \(K_{m,n}\) are \(H\)-\(V\)-super-strong-\((a,d)\)-antimagic decomposable with both \(m\) and \(n\) are even.

A Grundy \(k\)-coloring of a graph \(G\) is a proper \(k\)-coloring of vertices in \(G\) using colors \(\{1, 2, \cdots, k\}\) such that for any two colors \(x\) and \(y\), \(x<y\), any vertex colored \(y\) is adjacent to some vertex colored \(x\). The First-Fit or Grundy chromatic number (or simply Grundy number) of a graph \(G\), denoted by \(\Gamma \left(G\right)\), is the largest integer \(k\), such that there exists a Grundy \(k\)-coloring for \(G\). It can be easily seen that \(\Gamma \left(G\right)\) equals to the maximum number of colors used by the greedy (or First-Fit) coloring of \(G\). In this paper, we obtain the Grundy chromatic number of Cartesian Product of path graph, complete graph, cycle graph, complete graph, wheel graph and star graph.

Determining the Tutte polynomial \(T(G;x,y)\) of a graph network \(G\) is a challenging problem for mathematicians, physicians, and statisticians. This paper investigates a self-similar network model \(M(t)\) and derives its Tutte polynomial. In addition, we evaluate exact explicit formulas for the number of acyclic orientations and spanning trees of it as applications of the Tutte polynomial. Finally, we use the derived \(T(M(t);x,y)\) to obtain the Tutte polynomial of another self-similar model \(N(t)\) presented in [1] and correct the main result discussed in [1] by Ma et al. and test our result numerically by using Matlab.

A vertex-colouring of a graph \(\Gamma\) is rainbow vertex connected if every pair of vertices \((u,v)\) in \(\Gamma\) there is a \(u-v\) path whose internal vertices have different colours. The rainbow vertex connection number of a graph \(\Gamma\), is the minimum number of colours needed to make \(\Gamma\) rainbow vertex connected, denoted by \(rvc(\Gamma)\). Here, we study the rainbow vertex connection numbers of middle and total graphs. A total-colouring of a graph \(\Gamma\) is total rainbow connected if every pair of vertices \((u,v)\) in \(\Gamma\) there is a \(u-v\) path whose edges and internal vertices have different colours. The total rainbow connection number of \(\Gamma\), is the minimum number of colours required to colour the edges and vertices of \(\Gamma\) in order to make \(\Gamma\) total rainbow connected, denoted by \(trc(\Gamma)\). In this paper, we also research the total rainbow connection numbers of middle and total graphs.

The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of the weights \(\frac{2}{d_{u}+ d_{v}}\) of all edges \(uv\) of \(G\), where \(d_{u}\) denotes the degree of a vertex \(u\). Delorme et al. [1] (2002) put forward a conjecture concerning the minimum Randić index among all connected graphs with \(n\) vertices and the minimum degree at least \(k\). Motivated by this paper, a conjecture related to the minimum harmonic index among all connected graphs with \(n\) vertices and the minimum degree at least \(k\) was posed in [2]. In this work, we show that the conjecture is true for a connected graph on $n$ vertices with \(k\) vertices of degree \(n-1\), and it is also true for a \(k\)-tree. Moreover, we give a shorter proof of Liu’s result [3].

Let \(L\) be a unital ring with characteristic different from \(2\) and \(\mathcal{O}(L)\) be an algebra of Octonion over \(L\). In the present article, our attempt is to present the characterization as well as the matrix representation of some variants of derivations on \(\mathcal{O}(L)\). The matrix representation of Lie derivation of \(\mathcal{O}(L)\) and its decomposition in terms of Lie derivation and Jordan derivation of \(L\) and inner derivation of \(\mathcal{O}\) is presented. The result about the decomposition of Lie centralizer of \(\mathcal{O}\) in terms of Lie centralizer and Jordan centralizer of \(L\) is given. Moreover, the matrix representation of generalized Lie derivation (also known as \(D\)-Lie derivation) of \(\mathcal{O}(L)\) is computed.

A sum divisor cordial labeling of a graph \(G\) with vertex set \(V(G)\) is a bijection \(f\) from \(V(G)\) to \(\{1,2,\cdots,|V(G)|\}\) such that an edge \(uv\) is assigned the label \(1\) if \(2\) divides \(f(u)+f(v)\) and \(0\) otherwise; and the number of edges labeled with \(1\) and the number of edges labeled with \(0\) differ by at most \(1\). A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we discuss the sum divisor cordial labeling of transformed tree related graphs.

For a graph \(G\) and a positive integer \(k\), a royal \(k\)-edge coloring of \(G\) is an assignment of nonempty subsets of the set \(\{1, 2, \ldots, k\}\) to the edges of \(G\) that gives rise to a proper vertex coloring in which the color assigned to each vertex \(v\) is the union of the sets of colors of the edges incident with \(v\). If the resulting vertex coloring is vertex-distinguishing, then the edge coloring is a strong royal \(k\) coloring. The minimum positive integer \(k\) for which a graph has a strong royal \(k\)-coloring is the strong royal index of the graph. The primary emphasis here is on strong royal colorings of trees.

The coloring of all the edges of a graph \(G\) with the minimum number of colors, such that the adjacent edges are allotted a different color is known as the proper edge coloring. It is said to be equitable, if the number of edges in any two color classes differ by atmost one. In this paper, we obtain the equitable edge coloring of splitting graph of \(W_n\), \(DW_n\) and \(G_n\) by determining its edge chromatic number.