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.

Let \(G\) be a graph, and let \(a\) and \(b\) be integers with \(1 \leq a \leq b\). An \([a, b]\)-factor of \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(v \in V(G)\). In this paper, we obtain a sufficient condition for a graph to have \([a, b]\)-factors including given edges, extending a well-known sufficient condition for the existence of a \(k\)-factor.

We introduce the domination polynomial of a graph \(G\). The domination polynomial of a graph \(G\) of order \(n\) is defined as \(D(G, x) = \sum_{i=\gamma(G)}^{n} d(G, i)x^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\), and \(\gamma(G)\) is the domination number of \(G\). We obtain some properties of \(D(G, x)\) and its coefficients, and compute this polynomial for specific graphs.

For a tree \(T\), \(Leaf(T)\) denotes the set of leaves of \(T\), and \(T – Leaf(T)\) is called the stem of \(T\). For a graph \(G\) and a positive integer \(m\), \(\sigma_m(G)\) denotes the minimum degree sum of \(m\) independent vertices of \(G\). We prove the following theorem: Let \(G\) be a connected graph and \(k \geq 2\) be an integer. If \(\sigma_3(G) \geq |G| – 2k + 1\), then \(G\) has a spanning tree whose stem has at most \(k\) leaves.

A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most \(1\). The equitable chromatic threshold of a graph \(G\), denoted by \(\chi_m^*(G)\), is the minimum \(k\) such that \(G\) is equitably \(k’\)-colorable for all \(k’ > k\). Let \(G \times H\) denote the direct product of graphs \(G\) and \(H\). For \(n \geq m \geq 2\), we prove that \(\chi_m^*(K_m \times K_n)\) equals \(\left\lceil \frac{mn}{m+1} \right\rceil\) if \(n \equiv 2, \ldots, m \pmod{m+1}\), and equals \(m\left\lceil \frac{n}{s^*} \right\rceil\) if \(n \equiv 0, 1 \pmod{m+1}\), where \(s^*\) is the minimum positive integer such that \(s^* \nmid n\) and \(s^* \geq m+2\).