In this paper, we prove a fixed point theorem for weakly compatible mappings satisfying a general contractive condition of operator type. In short, we are going to study mappings \( A, B, S, T: X \to X \) for which there exists a right continuous function \( \psi: \mathbb{R}^+ \to \mathbb{R}^+ \) such that \(\psi(0) = 0\) and \(\psi(s)\leq s\) for \(s > 0.\) Moreover, for each \( x, y \in X \), one has \(O(f; d(Sx, Ty)) \leq \psi(O(f; M(x,y))),\) where \( O(f; \cdot) \) and \( f \) are defined in the first section. Also in the first section, we give some examples for \( O(f; \cdot) \). The second section contains the main result. In the last section, we give some corollaries and remarks.

We consider unitary graphs attached to \(\mathbb{Z}^{d}_{n}\) using an analogue of the Euclidean distance. These graphs are shown to be integral when \(d\) is odd or the dimension \(d\) is even.

A graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we determine the graph with the
largest spectral radius among all connected cactuses with n vertices and edge independence number \(q\).

In this study, we obtained lower and upper bounds for the Euclidean norm of a complex matrix \(A\) of order \(n \times n\). In addition,
we found lower and upper bounds for the spectral norms and Euclidean norms of the Hilbert matrix its Hadamard
square root, Cauchy-Toeplitz and Cauchy-Hankel matrices in the forms \(H = \left(\frac{1}{i + j – 1}\right)_{i,j=1}^n\),\(H^{\frac{01}{2}}=(\frac{1}{(i+j-1)}^{\frac{1}{2}})_{i,j=1}^n\); \(T_n = \left[\frac{1}{(g+(i + j)h)}_{i,j=1}^n\right]\), and \(H_n = \left[\frac{1}{(g+(i + j )h}\right]_{i,j=1}^n\), respectively.

Let \(G\) be a graph, and let \(a\) and \(b\) be nonnegative integers such that \(1 \leq a \leq b\). Let \(g\) and \(f\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(a \leq g(x) \leq f(x) \leq b\) for each \(x \in V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \((g, f)\)-factor if \(g(x) \leq d_G^h(x) \leq f(x)\) for all \(x \in V(G)\), where \(d_G^h(x) = \sum_{e \in E_x} h(e)\) is the fractional degree of \(x \in V(F)\) with \(E_x = \{e : e = xy \in E(G)\}\). The isolated toughness \(I(G)\) of a graph \(G\) is defined as follows: If \(G\) is a complete graph, then \(I(G) = +\infty\); else, \(I(G) = \min\{ \frac{|S|}{i(G-S)} : S \subseteq V(G), i(G – S) \geq 2 \}\), where \(i(G – S)\) denotes the number of isolated vertices in \(G – S\). In this paper, we prove that \(G\) has a fractional \((g, f)\)-factor if \(\delta(G) \geq I(G) \geq \frac{b(b-1)}{a}+1\). This result is best possible in some sense.

In this paper we prove that there exists one type of connected cubic graph,which minimizes the number of spanning trees over all other connected cubic graphs of the same order \(7\), \(n\geq 14\).

Let \(T = PSL(n, q)\) be a projective linear simple group, where \(n \geq 2\),\(q\) a prime power and \((n,q) \neq (2,2)\) and \((2,3)\). We classify all \(3— (v, k, 1)\) designs admitting an automorphism group \(G\) with \(T \unlhd G \leq Aut(T)\) and \(v=\frac{q^n-1}{q-1}.\)

In this paper, we introduce the notion of \(f\)-derivations and investigate the properties of \(f\)-derivations of lattice implication
algebras. We provide an equivalent condition for an isotone \(f\)-derivation in a lattice implication algebra. Additionally, we
characterize the fixed set \({Fix_d}(L)\) and \(\mathrm{Kerd}\) by \(f\)-derivations. Furthermore, we introduce
normal filters and obtain some properties of normal filters in lattice implication algebras.

We give a new combinatorial interpretation of Lah and \(r\)-Lah numbers.
We establish two cross recurrence relations: the first one, which uses
an algebraic approach, is a recurrence relation of order two with
rational coefficients; the second one uses a combinatorial proof and
is a recurrence relation with integer coefficients. We also express
\(r\)-Lah numbers in terms of Lah numbers. Finally, we give identities
related to rising and falling factorial powers.

In this paper, we reveal the yin-yang structure of the affine plane of order four by characterizing the unique blocking set as the
Mébius-Kantor configuration \(8_3\).