In this paper, we investigate some commutativity conditions and extend a remarkable result of Ram Awtar, when Lie ideal \(U\) becomes the part of the centre of \(M\) \(A\)-semiring \(R\).

A pebbling move involves removing two pebbles from one vertex and placing one on an adjacent vertex. The optimal pebbling number of a graph \(G\), denoted by \(f_{opt}(G)\), is the least positive integer \(n\) such that \(n\) pebbles are placed suitably on vertices of \(G\) and, for any specified vertex \(v\) of \(G\), one pebble can be moved to \(v\) through a sequence of pebbling moves. In this paper, we determine the optimal pebbling number of the square of paths and cycles.

In this paper, we verify the list edge coloring conjecture for pseudo- outerplanar graphs with maximum degree at least \(5\) and the equitable \(\Delta\)-coloring conjecture for all pseudo-outerplanar graphs.

We prove that the Cartesian product of two directed cycles of lengths \(n_1\) and \(n_2\) contains an antidirected Hamilton cycle, and hence is decomposable into antidirected Hamilton cycles, if and only if \(\gcd(n_1, n_2) = 2\). For the Cartesian product of \(k > 2\) directed cycles, we establish new sufficient conditions for the existence of an antidirected Hamilton cycle.

Let \(T\) be a tree with no vertices of degree \(2\) and at least one vertex of degree \(3\) or more. A Halin graph \(G\) is a plane graph obtained by connecting the leaves of \(T\) in the cyclic order determined by the planar drawing of \(T\). Let \(\Delta\), \(\lambda(G)\), and \(\chi(G^2)\) denote, respectively, the maximum degree, the \(L(2,1)\)-labeling number, and the chromatic number of the square of \(G\). In this paper, we prove the following results for any Halin graph \(G\): (1) \(\chi(G^2) \leq \Delta + 3\), and moreover \(\chi(G^2) = \Delta + 1\) if \(\Delta \geq 6\); (2) \(\lambda(G) \leq \Delta + 7\), and moreover \(\lambda(G) \leq \Delta + 2\) if \(\Delta \geq 9\).

In this paper, we investigate the zero divisor graph \(G_I(P)\) of a poset \(P\) with respect to a semi-ideal \(I\). We show that the girth of \(G_I(P)\) is \(3\), \(4\), or \(\infty\). In addition, it is shown that the diameter of such a graph is either \(1\), \(2\), or \(3\). Moreover, we investigate the properties of a cut vertex in \(G_I(P)\) and study the relation between semi-ideal \(I\) and the graph \(G_I(P)\), as established in (Theorem 3.9).

A graph \(G\) is \({super-connected}\), or \({super-\(\kappa\)}\), if every minimum vertex-cut isolates a vertex of \(G\). Similarly, \(G\) is \({super-restricted \;edge-connected}\), or \({super-\(\lambda’\)}\), if every minimum restricted edge-cut isolates an edge. We consider the total graph \(T(G)\) of \(G\), which is formed by combining the disjoint union of \(G\) and the line graph \(L(G)\) with the lines of the subdivision graph \(S(G)\); for each line \(l = (u,v)\) in \(G\), there are two lines in \(S(G)\), namely \((l,u)\) and \((l,v)\). In this paper, we prove that \(T(G)\) is super-\(\kappa\) if \(G\) is super-\(\kappa\) graph with \(\delta(G) \geq 4\). \(T(G)\) is super-\(\lambda’\) if \(G\) is \(k\)-regular with \(\kappa(G) \geq 3\). Furthermore, we provide examples demonstrating that these results are best possible.

The paper construct infinite classes of non-isomorphic \(3\)-connected simple graphs with the same total genus polynomial, using overlap matrix, symmetry and Gustin representation. This answers a problem (Problem \(3\) of Page \(38\)) of L.A. McGeoch in his PHD thesis.
The result is helpful for firms to make marketing decisions by calculating the graphs of user demand relationships of different complex ecosystems of platform products and comparing genus polynomials.

A necessary and sufficient condition of the complement to be cordial and its application are obtained.

In this paper, we introduce the notion of blockwise-bursts in array codes equippped with m-metric \([13]\) and obtain some bounds on the parameters of $m$-metric array codes for the detection and correction of blockwise-burst array errors.