The Laplacian-energy-like graph invariant of a graph \(G\), denoted by \(LEL(G)\), is defined as \(LEL(G) = \sum\limits_{i=1}^{n} \sqrt{\mu_i}\), where \(\mu_i\) are the Laplacian eigenvalues of graph \(G\). In this paper, we study the maximum \(LEL\) among graphs with a given number of vertices and matching number. Some results on \(LEL(G)\) and \(LEL(\overline{G})\) are obtained.

In this paper, we consider mixed arrangements, which are composed of
hyperplanes (or subspaces) and spheres. We investigate the posets of
their intersection sets and calculate the Möbius functions of the
mixed arrangements through the hyperplane (or subspace) arrangements’
Möbius functions. Furthermore, by employing the method of deletion
and restriction, we derive recursive formulas for the triples of
these mixed arrangements.

For every integer \(c\), let \(n = R_d(c)\) be the least integer such
that for every coloring \(\Delta: \{1, 2, \ldots, 2n\} \to \{0, 1\}\),
there exists a solution \((x_1, x_2, x_3)\) to
\[x_1 + x_2 + x_3 = c\]
such that \(x_i \neq x_j\) when \(i \neq j\),
and
\(\Delta(x_1) = \Delta(x_2) = \Delta(x_3)\).

In this paper, it is shown that for every integer \(c\),
\[R_d(c) =
\begin{cases}
4c + 8 & \text{if } c \geq 1,\\
8 & \text{if } -3 \leq c < -6,\\ 9 & \text{if} c=0,-2,-7,-8\\ 10 & \text{if } c =-1,-9 \\ |c| -\left\lfloor \frac{|c|-4}{5} \right\rceil & \text{if } c \leq -10. \end{cases}\]

A graph \(G\) with an even number of vertices is said to be
almost self-complementary if it is isomorphic to one of its
almost complements \(G^c – M\), where \(M\) denotes a perfect matching
in its complement \(G^c\). In this paper, we show that the diameter
of connected almost self-complementary graphs must be \(2\), \(3\), or
\(4\). Furthermore, we construct connected almost self-complementary
graphs with \(2n\) vertices having diameter \(3\) and \(4\) for each \(n \geq 3\),
and diameter \(2\) for each \(n \geq 4\), respectively. Additionally, we
also obtain that for any almost self-complementary graph \(G_n\) with
\(2n\) vertices, \(\lceil \sqrt{n}\rceil \leq \chi(G_n) \leq n\). By
construction, we verify that the upper bound is attainable for each
positive integer \(n\), as well as the lower bound when \(\sqrt{n}\)
is an integer.

A \(k\)-container \(C(u,v)\) in a graph \(G\) is a set of \(k\) internally
vertex-disjoint paths between vertices \(u\) and \(v\). A \(k^*\)-container
\(C(u,v)\) of \(G\) is a \(k\)-container such that \(C(u,v)\) contains all
vertices of \(G\). A graph is globally \(k^*\)-connected if there exists
a \(k^*\)-container \(C(u,v)\) between any two distinct vertices \(u\) and \(v\).
A \(k\)-regular graph \(G\) is super \(k\)-spanning connected if \(G\) is
\(i^*\)-connected for \(1 \leq i \leq k\). A graph \(G\) is \(1\)-fault-tolerant
Hamiltonian if \(G – F\) is Hamiltonian for any \(F \subseteq V(G)\) and
\(|F| = 1\). In this paper, we prove that for cubic graphs, every
super \(3\)-spanning connected graph is globally \(3^*\)-connected and
every globally \(3^*\)-connected graph is \(1\)-fault-tolerant Hamiltonian.
We present examples of super \(3\)-spanning connected graphs, globally
\(3^*\)-connected graphs that are not super \(3\)-spanning connected,
\(1\)-fault-tolerant Hamiltonian graphs that are globally \(1^*\)-connected
but not globally \(3^*\)-connected, and \(1\)-fault-tolerant Hamiltonian
graphs that are neither globally \(1^*\)-connected nor globally \(3^*\)-connected.
Furthermore, we prove that there are infinitely many graphs in each
such family.

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.