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 \(s^* \geq m+2\).

For an undirected graph \(G\) and a natural number \(n\), a \(G\)-design of order \(n\) is an edge partition of the complete graph \(K_n\) with \(n\) vertices into subgraphs \(G_1, G_2, \ldots\), each isomorphic to \(G\). A set \(T \subset V(K_n)\) is called a blocking set if it intersects the vertex set \(V(G_i)\) of each \(G_i\) in the decomposition but contains none of them. Extending previous work [J. Combin. Designs \(4 (1996), 135-142]\), where the authors proved that cycle designs admit no blocking sets, we establish that this result holds for all graphs \(G\). Furthermore, we show that for every graph \(G\) and every integer \(k \geq 2\), there exists a non-\(k\)-colorable \(G\)-design.

Let \(G\) be a planar graph with maximum degree \(\Delta(G)\). The least integer \(k\) such that \(G\) can be partitioned into \(k\) edge-disjoint forests, where each component is a path of length at most \(2\), is called the linear \(2\)-arboricity of \(G\), denoted by \(la_2(G)\). We establish new upper bounds for the linear \(2\)-arboricity of certain planar graphs.

A graph \(G\) of order \(n\) is called a bicyclic graph if \(G\) is connected and the number of edges of \(G\) is \(n+ 1\). In this paper, we study the lexicographic ordering of bicyclic graphs by spectral moments. For each of the three basic types of bicyclic graphs on a fixed number of vertices maximal and minimal graphs in the mentioned order are determined.