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.

Let us consider a~simple connected undirected graph \(G=(V,E)\). For a~graph \(G\) we define a~\(k\)-labeling \(\phi: V(G)\to \{1,2, \dots, k\}\) to be a~distance irregular vertex \(k\)-labeling of the graph \(G\) if for every two different vertices \(u\) and \(v\) of \(G\), one has \(wt(u) \ne wt(v),\) where the weight of a~vertex \(u\) in the labeling \(\phi\) is \(wt(u)=\sum\limits_{v\in N(u)}\phi(v),\) where \(N(u)\) is the set of neighbors of \(u\). The minimum \(k\) for which the graph \(G\) has a~distance irregular vertex \(k\)-labeling is known as distance irregularity strength of \(G,\) it is denoted as \(dis(G)\). In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order \(1,\) friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.

Elimination ideals are monomial ideals associated to simple graphs, not necessarily square–free, was introduced by Anwar and Khalid. These ideals are Borel type. In this paper, we obtain sharp combinatorial upper bounds of the Castelnuovo–Mumford regularity of elimination ideals corresponding to certain family of graphs.

Let \(G\) be a simple connected graph with vertex set \(V\) and diameter \(d\). An injective function \(c: V\rightarrow \{1,2,3,\ldots\}\) is called a radio labeling of \(G\) if \({|c(x) c(y)|+d(x,y)\geq d+1}\) for all distinct \(x,y\in V\), where \(d(x,y)\) is the distance between vertices \(x\) and \(y\). The largest number in the range of \(c\) is called the span of the labeling \(c\). The radio number of \(G\) is the minimum span taken over all radio labelings of \(G\). For a fixed vertex \(z\) of \(G\), the sequence \((l_1,l_2,\ldots,l_r)\) is called the level tuple of \(G\), where \(l_i\) is the number of vertices whose distance from \(z\) is \(i\). Let\(J^k(l_1,l_2,\ldots,l_r)\) be the wedge sum (i.e. one vertex union) of \(k\geq2\) graphs having same level tuple \((l_1,l_2,\ldots,l_r)\). Let \(J(\frac{l_1}{l’_1},\frac{l_2}{l’_2},\ldots,\frac{l_r} {l’_r})\) be the wedge sum of two graphs of same order, having level tuples  \((l_1,l_2,\ldots,l_r)\) and \((l’_1,l’_2,\ldots,l’_r)\). In this paper, we compute the radio number for some sub-families of \(J^k(l_1,l_2,\ldots,l_r)\) and \(J(\frac{l_1}{l’_1},\frac{l_2}{l’_2},\ldots,\frac{l_r}{l’_r})\).