A \({magic\; square}\) of order \(n\) is an \(n \times n\) array of integers from \(1, 2, \ldots, n^2\) such that the sum of the integers in each row, column, and diagonal is the same number. Two magic squares are \({equivalent}\) if one can be obtained from the other by rotation or reflection. The \({complement}\) of a magic square \(M\) of order \(n\) is obtained by replacing every entry \(a\) with \(n^2 + 1 – a\), yielding another magic square. A magic square is \({self-complementary}\) if it is equivalent to its complement. In this paper, we prove a structural theorem characterizing self-complementary magic squares and present a method for constructing self-complementary magic squares of even order. Combining this construction with the structural theorem and known results on magic squares, we establish the existence of self-complementary magic squares of order \(n\) for every \(n \geq 3\).

Let \(G\) be a graph on \(n\) vertices. If for any ordered set of vertices \(S = \{v_1, v_2, \ldots, v_k\}\), where the vertices in \(S\) appear in the sequence order \(v_1, v_2, \ldots, v_k\), there exists a \(v_1-v_k\) (Hamiltonian) path containing \(S\) in the given order, then \(G\) is \(k\)-ordered (Hamiltonian) connected. In this paper, we show that if \(G\) is \((k+1)\)-connected and \(k\)-ordered connected, then for any ordered set \(S\), there exists a \(v_1-v_k\) path \(P\) containing \(S\) in the given order such that \(|P| \geq \min\{n, \sigma_2(G) – 1\}\), where \(\sigma_2(G) = \min\{d_G(u) + d_G(v) : u,v \in V(G); uv \notin E(G)\}\) when \(G\) is not complete, and \(\sigma_2(G) = \infty\) otherwise. Our result generalizes several related results known before.

Let \(G\) be a simple graph. The incidence energy ( \(IE\) for short ) of \(G\) is defined as the sum of the singular values of the incidence matrix. In this paper, a new lower bound for \(IE\) of graphs in terms of the maximum degree is given. Meanwhile, an upper bound and a lower bound for \(IE\) of the subdivision graph and the total graph of a regular graph \(G\) are obtained, respectively.

The Hosoya polynomial of a graph \(G\) with vertex set \(V(G)\) is defined as \(H(G, z) = \sum_{u,v \in V(G)} x^{d_G(u,v)}\), where \(d_G(u,v)\) is the distance between vertices \(u\) and \(v\). A toroidal polyhex \(H(p,q,t)\) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, described by a string \((p,q,t)\) of three integers \((p \geq 2, q \geq 1, 0 \leq t \leq p-1)\). In this paper, we derive an analytical formula for calculating the Hosoya polynomial of \(H(p,q,t)\) for \(t = 0\) or \(p\leq 2q\) or \(p \leq q+t\). Notably, some earlier results in [2, 6, 26] are direct corollaries of our main findings.

Kotani and Sunada introduced the oriented line graph as a tool in the study of the Ihara zeta function of a finite graph. The spectral properties of the adjacency operator on the oriented line graph can be linked to the Ramanujan condition of the graph. Here, we present a partial characterization of oriented line graphs in terms of forbidden subgraphs. We also give a Whitney-type result, as a special case of a result by Balof and Storm, establishing that if two graphs have the same oriented line graph, they are isomorphic.

Let \(A\) be the \((0,1)\)-adjacency matrix of a simple graph \(G\), and \(D\) be the diagonal matrix \(diag(d_1, d_2, \ldots, d_n)\), where \(d_i\) is the degree of the vertex \(v_i\). The matrix \(Q(G) = D + A\) is called the signless Laplacian of \(G\). In this paper, we characterize the extremal graph for which the least signless Laplacian eigenvalue attains its minimum among all non-bipartite unicyclic graphs with given order and diameter.

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.