Let \(G=(V,\,E)\) be a simple graph with vertex set \(V(G)\) and edge set \(E(G)\). The Lanzhou index of a graph \(G\) is defined by \(Lz(G)=\sum\limits_{u \in V(G)} d_u^2\overline{d}_u\), where \(d_u\) (\(\overline{d}_u \) resp.) denotes the degree of the vertex \(u\) in \(G\) (\(\overline{G}\), the complement graph of \(G\) resp.). It has predictive powers to provide insights of chemical relevant properties of chemical graph structures. In this paper we discuss some properties of Lanzhou index. Several inequalities having lower and upper bound for the Lanzhou index in terms of first, second and third Zagreb indices, radius of graph, eccentric connectivity index, Schultz index, inverse sum indeg index and symmetric division deg index, are discussed. At the end the Lanzhou index of corona and join of graphs have been derived.

We define an extremal \((r|\chi)\)-graph as an \(r\)-regular graph with chromatic number \(\chi\) of minimum order. We show that the Turán graphs \(T_{ak,k}\), the antihole graphs and the graphs \(K_k\times K_2\) are extremal in this sense. We also study extremal Cayley \((r|\chi)\)-graphs and we exhibit several \((r|\chi)\)-graph constructions arising from Turán graphs.

A dominating broadcast of a graph \(G\) is a function \(f : V(G) \rightarrow \lbrace 0, 1, 2, \dots ,\text{diam}(G)\rbrace\) such that \(f(v) \leqslant e(v)\) for all \(v \in V(G)\), where \(e(v)\) is the eccentricity of \(v\), and for every vertex \(u \in V(G)\), there exists a vertex \(v\) with \(f(v) > 0\) and \(\text{d}(u,v) \leqslant f(v)\). The cost of \(f\) is \(\sum_{v \in V(G)} f(v)\). The minimum of costs over all the dominating broadcasts of \(G\) is called the broadcast domination number \(\gamma_{b}(G)\) of \(G\). A graph $G$ is said to be radial if \(\gamma_{b}(G)=\text{rad}(G)\). In this article, we give tight upper and lower bounds for the broadcast domination number of the line graph \(L(G)\) of \(G\), in terms of \(\gamma_{b}(G)\), and improve the upper bound of the same for the line graphs of trees. We present a necessary and sufficient condition for radial line graphs of central trees, and exhibit constructions of infinitely many central trees \(T\) for which \(L(T)\) is radial. We give a characterization for radial line graphs of trees, and show that the line graphs of the \(i\)-subdivision graph of \(K_{1,n}\) and a subclass of caterpillars are radial. Also, we show that \(\gamma_{b}(L(C))=\gamma(L(C))\) for any caterpillar \(C\).

In this paper we introduce the concept of independent fixed connected geodetic number and investigate its behaviours on some standard graphs. Lower and upper bounds are found for the above number and we characterize the suitable graphs achieving these bounds. We also define two new parameters connected geo-independent number and upper connected geo-independent number of a graph. Few characterization and realization results are formulated for the new parameters. Finally an open problem is posed.

Let \(E(H)\) and \(V(H)\) denote the edge set and the vertex set of the simple connected graph \(H\), respectively. The mixed metric dimension of the graph \(H\) is the graph invariant, which is the mixture of two important graph parameters, the edge metric dimension and the metric dimension. In this article, we compute the mixed metric dimension for the two families of the plane graphs viz., the Web graph \(\mathbb{W}_{n}\) and the Prism allied graph \(\mathbb{D}_{n}^{t}\). We show that the mixed metric dimension is non-constant unbounded for these two families of the plane graph. Moreover, for the Web graph \(\mathbb{W}_{n}\) and the Prism allied graph \(\mathbb{D}_{n}^{t}\), we unveil that the mixed metric basis set \(M_{G}^{m}\) is independent.

Consider a total labeling \(\xi\) of a graph \(G\). For every two different edges \(e\) and \(f\) of \(G\), let \(wt(e) \neq wt(f)\) where weight of \(e = xy\) is defined as \(wt(e)=|\xi(e) – \xi(x) – \xi(y)|\). Then \(\xi\) is called edge irregular total absolute difference \(k\)-labeling of \(G\). Let \(k\) be the minimum integer for which there is a graph \(G\) with edge irregular total absolute difference labeling. This \(k\) is called the total absolute difference edge irregularity strength of the graph \(G\), denoted \(tades(G)\). We compute \(tades\) of \(SC_{n}\), disjoint union of grid and zigzag graph.

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.