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 \(J(\frac{l_1}{l’_1},\frac{l_2}{l’_2},\ldots,\frac{l_r}{l’_r})\).

An antipodal labeling is a function \(f\ \)from the vertices of \(G\) to the set of natural numbers such that it satisfies the condition \(d(u,v) + \left| f(u) – f(v) \right| \geq d\), where d is the diameter of \(G\ \)and \(d(u,v)\) is the shortest distance between every pair of distinct vertices  \(u\) and \(v\) of \(G.\) The span of an antipodal labeling \(f\ \)is \(sp(f) = \max\{|f(u) – \ f\ (v)|:u,\ v\, \in \, V(G)\}.\) The antipodal number of~G, denoted by~an(G), is the minimum span of all antipodal labeling of~G. In this paper, we determine the antipodal number of Mongolian tent and Torus grid.

Let \( G \) be a \((p,q)\)-graph in which the edges are labeled \( k, k+1, \ldots, k+q-1 \), where \( k \geq 0 \). The vertex sum for a vertex \( v \) is the sum of the labels of the incident edges at \( v \). If the vertex sums are constant, modulo \( p \), then \( G \) is said to be \( k \)-edge-magic. In this paper, we investigate some classes of cubic graphs which are \( k \)-edge-magic. We also provide a counterexample to a conjecture that any cubic graph of order \( p \equiv 2 \pmod{4} \) is \( k \)-edge-magic for all \( k \).