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 \(8_3\).

A family of sets is called \(K\)-union distinct if all unions involving \(K\) or fewer members thereof are distinct. If a family of
sets is \(K\)-cover-free, then it is \(K\)-union distinct. In this paper, we recognize that this is only a sufficient condition and,
from this perspective, consider partially cover-free families of sets with a view to constructing union distinct families. The
role of orthogonal arrays and related combinatorial structures is explored in this context. The results are applied to find
efficient anti-collusion digital fingerprinting codes.

Let \(G\) be a \(2\)-edge-connected simple graph on \(n\) vertices, \(n \geq 3\). It is known that if \(G\) satisfies \(d(x) \geq \frac{n}{2}\) for every vertex \(x \in V(G)\), then \(G\) has a nowhere-zero \(3\)-flow, with several exceptions.In this paper, we prove that, with ten exceptions, all graphs with at most two vertices of degree less than \(\frac{n}{2}\) have nowhere-zero \(3\)-flows. More precisely, if \(G\) is a \(2\)-edge-connected graph on \(n\) vertices, \(n \geq 3\), in which at most two vertices have degree less than \(\frac{n}{2}\), then \(G\)
has a nowhere-zero \(3\)-flow if and only if \(G\) is not one of ten completely described graphs.