Let \(G = (V, E)\) be a connected graph. \(G\) is \({super-\lambda}\) if every minimum edge cut of \(G\) isolates a vertex. Moreover, an edge set \(S \subseteq E\) is a \({restricted\; edge\; cut}\) of \(G\) if \(G – S\) is disconnected and every component of \(G – S\) has at least \(2\) vertices. The \({restricted \;edge\; connectivity}\) of \(G\), denoted by \(\lambda'(G)\), is the minimum cardinality of all restricted edge cuts. Let \(\xi(G) = \min\{d_G(u) + d_G(v) – 2: uv \in E(G)\}\). We say \(G\) is \({\lambda’-optimal}\) if \(\lambda'(G) = \xi(G)\). In this paper, we provide a sufficient condition for bipartite graphs to be both super-\(\lambda\) and \(\lambda’\)-optimal.

The thickness \(\theta(G)\) of a graph \(G\) is the minimum number of planar spanning subgraphs into which \(G\) can be decomposed. In this note, we determine the thickness of the complete tripartite graph \(K_{l,m,n}\) (\(1 \leq m \leq n\)) for the following cases: (1) \(l + m \leq 5\); (2) \(l + m\) is even and \(n > \frac{1}{2}(l + m – 2)\); (3) \(l + m\) is odd and \(n > (l + m – 2)(l + m – 1)\).

We give an elementary, self-contained, and purely combinatorial proof of the Rayleigh monotonicity property of graphs.

Let \(D = (V, A)\) be a strongly connected digraph. \(D\) is called \(\lambda’\)-optimal if its restricted arc-connectivity equals the minimum arc degree. In this paper, we provide sufficient conditions for digraphs to be \(\lambda’\)-optimal.

In this paper, new families of Pell and Pell-Lucas numbers are introduced. In addition, we present the recurrence relations
and the generating functions of the new families for \(k = 2.\)

Consider a labeled and strongly oriented cycle \(\overrightarrow{C_m}\) and a set \(\mathcal{T} = \{\overrightarrow{C_n}, \overleftarrow{C_n}\}\), where \(\overrightarrow{C_n}\) and \(\overleftarrow{C_n}\) are two labeled and strongly oriented cycles with the same underlying graph and opposite orientations. Let \(h: E(\overrightarrow{C_m}) \to \Gamma\) be any function that sends every edge of \(\overrightarrow{C_m}\) to either \(\overrightarrow{C_n}\) or \(\overleftarrow{C_n}\). The primary goal of this paper is to study the underlying graph of the product \(\overrightarrow{C_m} \otimes_h \Gamma\), defined as follows:
\[ V(\overrightarrow{C_m} \otimes_h \Gamma) = V(\overrightarrow{C_m}) \times V(\overrightarrow{C_n}) \]
and
\[ ((a, b), (c, d)) \in E(\overrightarrow{C_m} \otimes_h \Gamma) \iff (a, c) \in E(\overrightarrow{C_m}) \wedge (b, d) \in h(a, c). \]
This product is of interest since it preserves various types of labelings, such as edge-magic and super edge-magic labelings. Additionally, we investigate the algorithmic complexity of determining whether a digraph \(\overrightarrow{D}\) can be factored using the product \(\otimes_h\), given a set of digraphs \(\Gamma\). This is the main topic of the third section of the paper.

Doubly resolvable \(2-(v,k,\lambda)\) designs \((DRDs)\) with small parameters and their resolutions which have orthogonal resolutions (\(RORs\)) are constructed and classified up to isomorphism. Exact values or lower bounds on the number of the nonisomorphic sets of \(7\) mutually orthogonal resolutions \((m-MORs)\) are presented. The implemented algorithms and the parameter range of this method are discussed, and the correctness of the computational results is checked in several ways.

Let \(G\) be a simple graph of order \(n\). We define a dominating set as a set \(S \subseteq V(G)\) such that every vertex of \(G\) is either in \(S\) or adjacent to a vertex in \(S\). The \({domination\; polynomial}\) of \(G\) is \(D(G, x) = \sum_{i=0}^{n} d(G, i)x^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\). Two graphs \(G\) and \(H\) are \({D-equivalent}\), denoted \(G \sim H\), if \(D(G, x) = D(H, x)\). The \({D-equivalence\; class}\) of \(G\) is \([G] = \{H \mid H \sim G\}\). Recently, determining the \(D\)-equivalence class of a given graph has garnered interest. In this paper, we show that for \(n \geq 3\), \([K_{n,n}]\) has size two. We conjecture that the complete bipartite graph \(K_{m,n}\) for \(m, n \geq 2\) is uniquely determined by its domination polynomial.

The Jacobi matrix polynomials and their orthogonality only for commutative matrices was first studied by Defez \(et. al\).
[Jacobi matrix differential equation, polynomial solutions and their properties. Comput. Math. Appl. \(48 (2004), 789-803]\). It is known that orthogonal matrix polynomials comprise an emerging field of study, with important results in both theory and applications continuing to appear in the literature. The main object of this paper is to derive various families of linear, multilateral and multilinear generating functions for the Jacobi matrix polynomials and the Gegenbauer matrix polynomials. Recurrence relations of Jacobi matrix polynomials are obtained. Some special cases of the results presented in this study are also indicated.

The positive index of inertia of a signed graph \(\Gamma\), denoted by \(,(\Gamma)\), is the number of positive eigenvalues of the adjacency matrix \(A(\Gamma)\) including multiplicities. In this paper we investigate the minimal positive index of inertia of
signed unicyclic graphs of order \(n\) with fixed girth and characterize the extremal graphs with the minimal positive index. Finally, we characterize the signed unicyclic graphs with the positive indices \(1\) and \(2\).