In this paper we study defensive alliances in some regular graphs. We determine which subgraphs could a critical defensive alliance of a graph \(G\) induce, if \(G\) is \(6\)-regular and the cardinality of the alliance is at most \(8\).

We present mean and non mean graphs of order \(\leq 6\), and give an upper bound for the number of edges of a graph with certain number of vertices to be a mean graph, and we show that the maximum vertex degree could be found in mean graphs depending on the number of edges. Also, we construct families of mean graphs depending on other mean and non mean graphs.

Let \(G = (V, E)\) be a finite, simple, and undirected graph of order \(p\) and size \(q\). A super edge-magic total labeling of a graph \(G\) is a bijection \(\lambda: V(G) \cup E(G) \rightarrow \{1, 2, \ldots, p + q\}\), where vertices are labeled with \(1, 2, \ldots, p\) and there exists a constant \(t\) such that \(f(x) + f(xy) +f(y) = t\), for every edge \(xy \in E(G)\). The super edge-magic deficiency of a graph \(G\), denoted by \(\mu_s(G)\), is the minimum nonnegative integer \(n\) such that \(G \cup nK_1\) has a super edge-magic total labeling, or \(\infty\) if no such \(n\) exists. In this paper, we investigate the super edge-magic deficiency of a forest consisting of stars.

The paper begins with a simple circular lock problem that shows how the Combinatorial Nullstellensatz relates to the discrete Fourier Transform.Specifically, the lock shows a relationship between detecting perfect matchings in bipartite graphs using the Combinatorial Nullstellensatz and detecting a maximum rank independent set in the intersection of two matroids in the Fourier transform of a specially chosen function. Finally, an application of the uncertainity principle computes a lower bound for the product of perfect matchings and the number of independent sets.

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.