For any vertex \(x\) in a connected graph \(G\) of order \(n \geq 2\), a set \(S \subseteq V(G)\) is a \(z\)-detour monophonic set of \(G\) if each vertex \(v \in V(G)\) lies on a \(x-y\) detour monophonic path for some element \(y \in S\). The minimum cardinality of a \(x\)-detour monophonic set of \(G\) is the \(x\)-detour monophonic number of \(G\), denoted by \(dm_z(G)\). An \(x\)-detour monophonic set \(S_x\) of \(G\) is called a minimal \(x\)-detour monophonic set if no proper subset of \(S_x\) is an \(x\)-detour monophonic set. The upper \(x\)-detour monophonic number of \(G\), denoted by \(dm^+_x(G)\), is defined as the maximum cardinality of a minimal \(x\)-detour monophonic set of \(G\). We determine bounds for it and find the same for some special classes of graphs. For positive integers \(r, d,\) and \(k\) with \(2 \leq r \leq d\) and \(k \geq 2\), there exists a connected graph \(G\) with monophonic radius \(r\), monophonic diameter \(d\), and upper \(z\)-detour monophonic number \(k\) for some vertex \(x\) in \(G\). Also, it is shown that for positive integers \(j, k, l,\) and \(n\) with \(2 \leq j \leq k \leq l \leq n – 7\), there exists a connected graph \(G\) of order \(n\) with \(dm_x(G) = j\), \(dm^+_x(G) = l\), and a minimal \(x\)-detour monophonic set of cardinality \(k\).

Many authors define certain generalizations of the usual Fibonacci, Pell, and Lucas numbers by matrix methods and then obtain the Binet formulas and combinatorial representations of the generalizations of these number sequences. In this article, we firstly define and study the generalized Gaussian Fibonacci numbers and then find the matrix representation of the generalized Gaussian Fibonacci numbers and prove some theorems by these matrix representations.

Given two sets \(A, B \subset \mathbb{F}_q\), of elements of the finite field \(\mathbb{F}_q\), of \(q\) elements, Shparlinski (2008) showed that the product set \(\mathcal{AB} = \{ab \mid a \in \mathcal{A}, b \in \mathcal{B}\}\) contains an arithmetic progression of length \(k \geq 3\) provided that \(k

3\) is the characteristic of \(\mathbb{F}\), and \(|\mathcal{A}||\mathcal{B}| \geq 2q^{2-1/(k-1)}\). In this paper, we recover Shparlinski’s result for the case of 3-term arithmetic progressions via spectra of product graphs over finite fields. We also illustrate our method in the setting of residue rings. Let \(m\) be a large integer and \(\mathbb{Z}/m\mathbb{Z}\) be the ring of residues mod \(m\). For any two sets \(\mathcal{A}, \mathcal{B} \subset \mathbb{Z}/m\mathbb{Z}\) of cardinality \[|\mathcal{A}||\mathcal{B}| > m(\frac{r(m)m}{r(m)^{\frac{1}{2}} + 1})\], the product set \(\mathcal{AB}\) contains a \(3\)-term arithmetic progression, where \(r(m)\) is the smallest prime divisor of \(m\) and \(r(m)\) is the number of divisors of \(m\). The spectral proofs presented in this paper avoid the use of character and exponential sums, the usual tool to deal with problems of this kind.

A proper edge-coloring of a graph \(G\) with colors \(1, \ldots, t\) is called an interval \(t\)-coloring if the colors of edges incident to any vertex of \(G\) form an interval of integers. A graph \(G\) is interval colorable if it has an interval \(t\)-coloring for some positive integer \(t\). For an interval colorable graph \(G\), the least value of \(t\) for which \(G\) has an interval \(t\)-coloring is denoted by \(w(G)\). A graph \(G\) is outerplanar if it can be embedded in the plane so that all its vertices lie on the same (unbounded) face. In this paper, we show that if \(G\) is a 2-connected outerplanar graph with \(\Delta(G) = 3\), then \(G\) is interval colorable and \[ w(G) = \begin{cases} 3, & \text{if } |V(G)| \text{ is even}, \\ 4, & \text{if } |V(G)| \text{ is odd}. \end{cases} \]
We also give a negative answer to the question of Axenovich on the outerplanar triangulations.

In this paper, we characterize all finite abelian groups with isomorphic intersection graphs. This solves a conjecture proposed by \(B\).Zelinka.

This paper devotes to solving the following conjecture proposed by Gvozdjak: “An \((a, b; n)\)-graceful labeling of \(P_n\) exists if and only if the integers \(a, b, n\) satisfy (1) \(b – a\) has the same parity as \(n(n + 1)/2\); (2) \(0 < |b – a| \leq (n + 1)/2\) and (3) \(n/2 \leq a + b \leq 3n/2\).'' Its solving can shed some new light on solving the famous Oberwolfach problem. It is shown that the conjecture is true for every \(n\) if the conjecture is true when \(n \leq 4a + 1\) and \(a\) is a fixed value. Moreover, we prove that the conjecture is true for \(a = 0, 1, 2, 3, 4, 5, 6\).

The aim of this paper is to show that the corona \(P_n \bigodot P_m\) between two paths \(P_n\) and \(P_m\) is cordial for all \(n \geq 1\) and \(m \geq 1\). Also, we prove that except for \(n\) and \(m\) being congruent to \(2 \pmod{4}\), the corona \(C_n \bigodot C_m\) between two cycles \(C_n\) and \(C_m\) is cordial. Furthermore, we show that if \(n \equiv 2 \pmod{4}\) and \(m\) is odd, then \(C_n \bigodot C_m\) is not cordial.

In this paper, we establish some general identities involving the weighted row sums of a Riordan array and hyperharmonic numbers. From these general identities, we deduce some particular identities involving other special combinatorial sequences, such as the Stirling numbers, the ordered Bell numbers, the Fibonacci numbers, the Lucas numbers, and the binomial coefficients.

In this paper, we consider the relationship between toughness and the existence of \([a, b]\)-factors with inclusion/exclusion properties. We obtain that if \(t(G) \geq a – 1 + \frac{a – 1}{b}\) with \(b > a > 2\), where \(a, b\) are two integers, then for any two given edges \(e_1\) and \(e_2\), there exist an \([a, b]\)-factor including \(e_1, e_2\); and an \([a, b]\)-factor including \(e_1\) and excluding \(e_2\); as well as an \((a, b)\)-factor excluding \(e_1, e_2\). Furthermore, it is shown that the results are best possible in some sense.

In this paper, we will determine the NBB bases with respect to a certain standard ordering of atoms of lattices of \(321\)-\(312\)-\(231\)-avoiding permutations and of \(321\)-avoiding permutations with the weak Bruhat order. Using our expressions of NBB bases, we will calculate the Möbius numbers of these lattices. These values are shown to be related to Fibonacci polynomials.