A chemical structure specifies the molecular geometry of a given molecule or solid in the form of atom arrangements. One way to analyze its properties is to simulate its formation as a product of two or more simpler graphs. In this article, we take this idea to find upper and lower bounds for the generalized Randić index \(\mathcal{R}_{\alpha}\) of four types of graph products, using combinatorial inequalities. We finish this paper by providing the bounds for \(\mathcal{R}_{\alpha}\) of a line graph and rooted product of graphs.

Let \(G\) be a \((p, q)\) graph. Let \(f: V(G) \to \{1, 2, \ldots, k\}\) be a map where \(k \in \mathbb{N}\) is a variable and \(k > 1\). For each edge \(uv\), assign the label \(\gcd(f(u), f(v))\). \(f\) is called \(k\)-Total prime cordial labeling of \(G\) if \(\left|t_{f}(i) – t_{f}(j)\right| \leq 1\), \(i, j \in \{1, 2, \ldots, k\}\) where \(t_{f}(x)\) denotes the total number of vertices and edges labeled with \(x\). A graph with a \(k\)-total prime cordial labeling is called \(k\)-total prime cordial graph. In this paper, we investigate the 4-total prime cordial labeling of some graphs like dragon, Möbius ladder, and corona of some graphs.

Let \(G = (V, E)\) be a graph with vertex set \(V\) and edge set \(E\). An edge labeling \(f: E \to Z_{2}\) induces a vertex labeling \(f^{+} : V \to Z_{2}\) defined by \( f^{+}(v) \equiv \sum_{uv \in E} f(uv) \pmod 2 \), for each vertex \(v \in V\). For \(i \in Z_{2}\), let \( v_{f}(i) = |\{v \in V : f^+(v) = i\}| \) and \( e_{f}(i) = |\{e \in E : f(e) = i\}| \). An edge labeling \(f\) of a graph \(G\) is said to be edge-friendly if \( |e_{f}(1) – e_{f}(0)| \le 1 \). The set \(\{v_f(1) – v_f(0) : f \text{ is an edge-friendly labeling of } G\}\) is called the full edge-friendly index set of \(G\). In this paper, we shall determine the full edge-friendly index sets of one point union of cycles.

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

A graph \(G\) is called a fractional ID-\((g,f)\)-factor-critical covered graph if for any independent set \(I\) of \(G\) and for every edge \(e \in E(G-I)\), \(G-I\) has a fractional \((g,f)\)-factor \(h\) such that \(h(e) = 1\). We give a sufficient condition using degree condition for a graph to be a fractional ID-\((g,f)\)-factor-critical covered graph. Our main result is an extension of Zhou, Bian, and Wu’s previous result [S. Zhou, Q. Bian, J. Wu, A result on fractional ID-\(k\)-factor-critical graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 87(2013) 229–236] and Yashima’s previous result [T. Yashima, A degree condition for graphs to be fractional ID-\([a,b]\)-factor-critical, Australasian Journal of Combinatorics 65(2016) 191–199].