A graph \(G\) is said to be equitably \(k\)-colorable if the vertex set of \(G\) can be divided into \(k\) independent sets for which any two sets differ in size at most one. The equitable chromatic number of \(G\), \(\chi_{=}(G)\), is the minimum \(k\) for which \(G\) is equitably \(k\)-colorable. The equitable chromatic threshold of \(G\), \(\chi_m^*(G)\), is the minimum \(k\) for which \(G\) is equitably \(k’\)-colorable for all \(k’ \geq k\). In this paper, the exact values of \(\chi_m^*(P_{n’,2} \square K_{m,n})\) and \(\chi_{=}(P_{n’,m} \square K_{m,n})\) are obtained except that \(3 \leq \xi_m^*(P_{5,2} \square K_{m,n}) = \chi_{=}(P_{s,m} \square K_{m,n}) \leq 4\) when \(m+n \geq 3\min\{m,n\} + 2\) or \(m+n < 3\min\{m,n\} – 2\).

The sum-connectivity energy of a graph is defined as the sum of the absolute value of all the eigenvalues of its sum-connectivity matrix. In this paper, we give further lower and upper bounds for the sum-connectivity energy in terms of the number of vertices, number of edges, the harmonic index, and determinant of the sum-connectivity matrix. We also show that among connected graphs with \(n\) vertices, the star graph \(K_{1,n-1}\) has the minimum sum-connectivity energy.

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, \(4\)-valent one-regular graphs of order \(5p^2\), where \(p\) is a prime, are classified.

In this paper, we obtain that the characteristic polynomials of the signless Laplacian matrix of \(Q(G)\), \(R(G)\), \(T(G)\) can be expressed in terms of the characteristic polynomial of \(G\) when \(G\) is a regular or semiregular graph, from which upper bounds for the incidence energy of \(Q(G)\), \(R(G)\), \(T(G)\) are deduced.

Let \(R\) be a commutative ring with unity. The co-maximal ideal graph of \(R\), denoted by \(\Gamma(R)\), is a graph whose vertices are the proper ideals of \(R\) which are not contained in the Jacobson radical of \(R\), and two vertices \(I_1\) and \(I_2\) are adjacent if and only if \(I_1 + I_2 = R\). We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012, the following question was posed: If \(\Gamma(R)\) is an infinite star graph, can \(R\) be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question.

In this paper, a generalization of the Stirling numbers of the first and second kind, called \(m\) -Stirling numbers of the first and second kind, are derived. Based on the colored base-\(m\) number system, we give a combinatorial interpretation of \(m\) -Stirling numbers of the second kind. Some basic properties of the two kinds of \(m\) -Stirling numbers, including generating functions, explicit expressions, and recurrence relations, are also obtained.

A proper \(k\)-total coloring of a simple graph \(G\) is called \(k\)-vertex-distinguishing proper total coloring (\(k\)-VDTC) if for any two distinct vertices \(u\) and \(v\) of \(G\), the set of colors assigned to \(u\) and its incident edges differs from the set of colors assigned to \(v\) and its incident edges. The minimum number of colors required for a vertex-distinguishing proper total coloring of \(G\), denoted by \(\chi_{vt}(G)\), is called the vertex-distinguishing proper total chromatic number. For \(p\) even, \(p \geq 4\) and \(q \geq 3\), we will obtain vertex-distinguishing proper total chromatic numbers of complete \(p\)-partite graphs with each part of cardinality \(q\).

Let \(G\) be a simple graph of order \(n\). The domination polynomial of \(G\) is the polynomial \(D(G, x) = \sum_{i=0}^{n} d(G, i)\lambda^i\), where \(d(G, i)\) is the number of dominating sets of \(G\) of size \(i\). Every root of \(D(G, \lambda)\) is called a domination root of \(G\). It is clear that \((0, \infty)\) is a zero-free interval for the domination polynomial of a graph. It is interesting to investigate graphs that have complex domination roots with positive real parts. In this paper, we first investigate the complexity of the domination polynomial at specific points. Then, we present and investigate some families of graphs whose complex domination roots have positive real parts.

A quasi-tree is a graph for which the deletion of some vertex results in a tree. We determine the unique graph with minimum distance spectral radius among quasi-trees with fixed order and the unique graph with maximum distance spectral radius among cycle-containing quasi-trees with fixed order.

We initiate the study of double outer-independent domination in graphs. A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph \(G\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) is dominated by at least two vertices of \(D\), and the set \(V(G) \setminus D\) is independent. The double outer-independent domination number of a graph \(G\) is the minimum cardinality of a double outer-independent dominating set of \(G\). First, we discuss the basic properties of double outer-independent domination in graphs. We find the double outer-independent domination numbers for several classes of graphs. Next, we prove lower and upper bounds on the double outer-independent domination number of a graph, and we characterize the extremal graphs. Then, we study the influence of removing or adding vertices and edges. We also give Nordhaus-Gaddum type inequalities.

For any two graphs \(F_1\) and \(F_2\), the graph Ramsey number \(r(F_1, F_2)\) is the smallest positive integer \(N\) with the property that every graph of at least \(N\) vertices contains \(F_1\) or its complement contains \(F_2\) as a subgraph. In this paper, we consider the Ramsey numbers for theta-complete graphs. In fact, we prove that \(r(\theta_n, K_5) = 4n-3\) for \(n \geq 6\) and \(n \geq 10\).

The Randić index \(R\) is an important topological index in chemistry. In order to attack some conjectures concerning the Randić index, a modification \(R’\) of this index was introduced by Dvorak et al. [6]. The \(R’\) index of a graph \(G\) is defined as the sum of the weights \(\frac{1}{\max\{{d(u)d(v)}\}}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). We first give a best possible lower bound of \(R’\) for a graph with minimum degree at least two and characterize the corresponding extremal graphs, and then we establish some relations between \(R’\) and the chromatic number, the girth of a graph.

There are operations that transform a map \(\mathcal{M}\) (an embedding of a graph on a surface) into another map on the same surface, modifying its structure and consequently its set of flags \(\mathcal{F(M)}\). For instance, by truncating all the vertices of a map \(\mathcal{M}\), each flag in \(\mathcal{F(M)}\) is divided into three flags of the truncated map. Orbanić, Pellicer, and Weiss studied the truncation of \(k\)-orbit maps for \(k \leq 3\). They introduced the notion of \(T\)-compatible maps in order to give a necessary condition for a truncation of a \(k\)-orbit map to be either \(k\)-, \(\frac{3k}{2}\)-, or \(3k\)-orbit map. Using a similar notion, by introducing an appropriate partition on the set of flags of the maps, we extend the results on truncation of \(k\)-orbit maps for \(k \leq 7\) and \(k = 9\).

Let \(\Re_\beta\) denote the set of trees on \(n = kG + 1\) (\(k \geq 2\)) vertices with matching number \(\beta\). In this paper, the trees with minimal spectral radius among \(\Re_\beta\) (\(2 \leq \delta \leq 4\)) are determined, respectively.

Generalized whist tournament designs and ordered whist tournament designs are relatively new specializations of whist tournament designs, having first appeared in \(2003\) and \(1996\), respectively. In this paper, we extend the concept of an ordered whist tournament to a generalized whist tournament and introduce an entirely new combinatorial design, which we call a generalized ordered whist tournament. We focus specifically on generalized whist tournaments for games of size \(6\) and teams of size \(3\), where the number of players is a prime of the form \(6n+1\), and prove that these tournaments exist for all primes \(p\) of the form \(p=6n+1\), with the possible exception of \(p \in \{7, 13, 19, 37, 61, 67\}\).

A regular graph \(\Gamma\) is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. Some authors classified semisymmetric cubic graphs of orders \(10p\) and \(10p^2\). Also, it is proved that there is no connected semisymmetric cubic graph of order \(10p^3\). In this paper, we continue this work and prove that there is no connected semisymmetric cubic graph of order \(10p^n\), where \(n \geq 4\), \(p \geq 7\), and \(p \neq 11\).

In this.paper, by joint tree model, we obtain the genera of two types of graphs, which are suspensions of cartesian products of two types of bipartite graphs from a vertex.

Let \(G\) be a connected graph with a perfect matching on \(2n\) vertices (\(n \geq 2\)). A graph \(G’\) is a contraction of \(G\) if it can be obtained from \(G\) by a sequence of edge contractions. Then \(G\) is said to be edge contractible if for any contraction \(G’\) of \(G\) with \(|V(G’)|\) even, \(G’\) has a perfect matching. In this note, we obtain a sufficient and necessary condition for a graph to be an edge contractible graph.

All finite Jacobson graphs with a Hamiltonian cycle or path, or Eulerian tour or trail are determined, and it is shown that a finite Jacobson graph is Hamiltonian if and only if it is pancyclic. Also, the length of the longest induced cycles and paths in finite Jacobson graphs are obtained.

A vertex subset \(S\) of a digraph \(D\) is called a dominating set of \(D\) if every vertex not in \(S\) is adjacent from at least one vertex in \(S\). The domination number of \(D\), denoted by \(\gamma(D)\), is the minimum cardinality of a dominating set of \(D\). We characterize the rooted trees and connected contrafunctional digraphs \(D\) of order \(n\) satisfying \(\gamma(D) = \left\lceil \frac{n}{2}\right\rceil\). Moreover, we show that for every digraph \(D\) of order \(n\) with minimum in-degree at least one, \(\gamma(D) \leq \frac{(k+1)n}{2k+1}\), where \(2k+1\) is the length of a shortest odd directed cycle in \(D\), and we characterize the corresponding digraphs achieving this upper bound. In particular, if \(D\) contains no odd directed cycles, then \(\gamma(D) \leq \frac{n}{2}\).

A graph is called degree-magic if it admits a labelling of the edges by integers \(\{1, 2, \ldots, |E(G)|\}\) such that the sum of the labels of the edges incident with any vertex \(v\) is equal to \(\left(1 + |E(G)|\right)/2 \deg(v)\). In this paper, we show that a class of join graphs are degree-magic.

A vertex-deleted unlabeled subgraph of a graph \(G\) is called a card of \(G\). A card of \(G\) with which the degree of the deleted vertex is also given is called a degree-associated card or dacard of \(G\). The degree-associated reconstruction number, \(\mathrm{drn}(G)\), of a graph \(G\) is the size of the smallest collection of dacards of \(G\) that uniquely determines \(G\). The maximal subgraph without end vertices of a graph \(G\) that is not a tree is called the pruned graph of \(G\). It is shown that \(\mathrm{drn}\) of some connected graphs with regular pruned graph is \(2\) or \(3\).

The Wiener index of a connected graph is the sum of distances between all pairs of vertices in the graph. Feng et al. in [The hyper-Wiener index of bicyclic graphs, Utilitas Math., \(84(2011) 97-104\)] determined the bicyclic graphs having the largest Wiener index. In this article, we determine the graphs having the second up to seventh largest Wiener indices among all bicyclic graphs with \(n\) vertices.

The matching energy of a graph was introduced by Gutman and Wagner in \(2012\) and defined as the sum of the absolute values of zeros of its matching polynomial. In this paper, we completely determine the graph with minimum matching energy in tricyclic graphs with given girth and without \(K_4\)-subdivision.

In this paper, we define and study the Gaussian Fibonacci and Gaussian Lucas \(p\)-numbers. We give generating functions, Binet formulas, explicit formulas, matrix representations, and sums of Gaussian Fibonacci \(p\)-numbers by matrix methods. For \(p = 1\), these Gaussian Fibonacci and Gaussian Lucas \(p\)-numbers reduce to the Gaussian Fibonacci and the Gaussian Lucas numbers.

Let \(G\) be a graph of order \(n\) and let \(Q(G, x) = \det(xI – Q(G)) = \sum_{i=0}^{n}(-1)^i\zeta_i(G)x^{n-i}\) be the characteristic polynomial of the signless Laplacian matrix of \(G\). We show that the Lollipop graph, \(L_{n,3}\), has the maximal \(Q\)-coefficients, among all unicyclic graphs of order \(n\) except \(C_n\). Moreover, we determine graphs with minimal \(Q\)-coefficients, among all unicyclic graphs of order \(n\).

Let \(G\) be a graph with \(n\) vertices, \(\mathcal{G}(G)\) the subdivision graph of \(G\). \(V(G)\) denotes the set of original vertices of \(G\). The generalized subdivision corona vertex graph of \(G\) and \(H_1, H_2, \ldots, H_n\) is the graph obtained from \(\mathcal{G}(G)\) and \(H_1, H_2, \ldots, H_n\) by joining the \(i\)th vertex of \(V(G)\) to every vertex of \(H_i\). In this paper, we determine the Laplacian (respectively, the signless Laplacian) characteristic polynomial of the generalized subdivision corona vertex graph. As an application, we construct infinitely many pairs of cospectral graphs.

In the paper, we show that the orientable genus of the generalized Petersen graph \(P(km, m)\) is at least \( \frac{km}{4} – \frac{m}{2}-\frac{km}{4m-4}+1\) if \(m\geq 4\) and \(k \geq 3\). We determine the orientable genera of \(P(3m, m)\), \(P(4k, 4)\), \(P(4m, m)\) if \(m \geq 4\), \(P(6m, m)\) if \(m \equiv 0 \pmod{2}\) and \(m \geq 6\), and so on.

Assume that \(\mu_1, \mu_2, \ldots, \mu_n\) are the eigenvalues of the Laplacian matrix of a graph \(G\). The Laplacian Estrada index of \(G\), denoted by \(LEE(G)\), is defined as \(LEE(G) = \sum_{i=1}^{n} e^{\mu_i}\). In this note, we give an upper bound on \(LEE(G)\) in terms of chromatic number and characterize the corresponding extremal graph.

In this note, we provide a combinatorial proof of a recent formula for the total number of peaks and valleys (either strict or weak) within the set of all compositions of a positive integer into a fixed number of parts.

The adjacent vertex distinguishing total chromatic number \(\chi_{at}(G)\) of a graph \(G\) is the smallest integer \(k\) for which \(G\) admits a proper \(k\)-total coloring such that no pair of adjacent vertices are incident to the same set of colors. Snarks are connected bridgeless cubic graphs with chromatic index \(4\). In this paper, we show that \(\chi_{at}(G) = 5\) for two infinite subfamilies of snarks, i.e., the Loupekhine snark and Blanusa snark of first and second kind. In addition, we give an adjacent vertex distinguishing total coloring using \(5\) colors for Watkins snark and Szekeres snark, respectively.

Let \(G\) be a tricyclic graph. Tricyclic graphs are connected graphs in which the number of edges equals the number of vertices plus two. In this paper, we determine graphs with the largest signless Laplacian spectral radius among all the tricyclic graphs with \(n\) vertices and diameter \(d\).

A pebbling move on a graph \(G\) consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number of a graph \(G\), denoted by \(f(G)\), is the least integer \(n\) such that, however \(n\) pebbles are located on the vertices of \(G\), we can move one pebble to any vertex by a sequence of pebbling moves. For any connected graphs \(G\) and \(H\), Graham conjectured that \(f(G \times H) \leq f(G)f(H)\). In this paper, we give the pebbling number of some graphs and prove that Graham’s conjecture holds for the middle graphs of some even cycles.

Graph embedding is an important factor to evaluate the quality of an interconnection network. It is also a powerful tool for implementation of parallel algorithms and simulation of different interconnection networks. In this paper, we compute the exact wirelength of embedding circulant networks into cycle-of-ladders.

In this paper, we characterize the extremal digraph with the maximal signless Laplacian spectral radius and the minimal distance signless Laplacian spectral radius among all simple connected digraphs with a given dichromatic number, respectively.

Given a graph \(G = (V, E)\) with no isolated vertex, a subset \(S \subseteq V\) is a total dominating set of \(G\) if every vertex in \(V\) is adjacent to a vertex in \(S\). A total dominating set \(S\) of \(G\) is a locating-total dominating set if for every pair of distinct vertices \(u\) and \(v\) in \(V – S\), we have \(N(u) \cap S \neq N(v) \cap S\), and \(S\) is a differentiating-total dominating set if for every pair of distinct vertices \(u\) and \(v\) in \(V\), we have \(N(u) \cap S \neq N(v) \cap S\). The locating-total domination number (or the differentiating-total domination number) of \(G\), denoted by \(\gamma_t^L(G)\) (or \(\gamma_t^D(G)\)), is the minimum cardinality of a locating-total dominating set (or a differentiating-total dominating set) of \(G\). In this paper, we investigate the bounds of locating and differentiating-total domination numbers of unicyclic graphs.

Motzkin posed the problem of finding the maximal density \(\mu(M)\) of sets of integers in which the differences given by a set \(M\) do not occur. The problem is already settled when \(|M| \leq 2\) or \(M\) is a finite arithmetic progression. In this paper, we determine \(\mu(M)\) when \(M\) has some other structure. For example, we determine \(\mu(M)\) when \(M\) is a finite geometric progression.

For vertices \(u, v\) in a connected graph \(G\), a \(u-v\) chordless path in \(G\) is a \(u-v\) monophonic path. The monophonic interval \(J_G[u, v]\) consists of all vertices lying on some \(u-v\) monophonic path in \(G\). For \(S \subseteq V(G)\), the set \(J_G[S]\) is the union of all sets \(J_G[u, v]\) for \(u, v \in S\). A set \(S \subseteq V(G)\) is a monophonic set of \(G\) if \(J_G[S] = V(G)\). The cardinality of a minimum monophonic set of \(G\) is the monophonic number of \(G\), denoted by \(mn(G)\). In this paper, bounds for the monophonic number of the strong product graphs are obtained, and for several classes, improved bounds and exact values are obtained.

A hypergraph is a useful tool to model complex systems and can be considered a natural generalization of graphs. In this paper, we define some operations of fuzzy hypergraphs and strong fuzzy \(r\)-uniform hypergraphs, such as Cartesian product, strong product, normal product, lexicographic product, union, and join. We prove that if a hypergraph \(H\) is formed by one of these operations, then this hypergraph is a fuzzy hypergraph or a strong fuzzy \(r\)-uniform hypergraph. Finally, we discuss an application of fuzzy hypergraphs.

Let \(p_e(n)\) be the number of ways to make change for \(n\) cents using pennies, nickels, dimes, and quarters. By manipulating the generating function for \(p_e(n)\), we prove that the sequence \(\{p_e(n) \pmod{\ell^j}\}\) is periodic for every prime power \(\ell\).

In 1972, Chvatal and Erdős showed that the graph \(G\) with independence number \(\alpha(G)\) no more than its connectivity \(\kappa(G)\) (i.e., \(\kappa(G) \geq \alpha(G)\)) is hamiltonian. In this paper, we consider a kind of Chvatal and Erdős type condition on edge-connectivity \(\lambda(G)\) and matching number (edge independence number). We show that if \(\lambda(G) \geq \alpha'(G) – 1\), then \(G\) is either supereulerian or in a well-defined family of graphs. Moreover, we weaken the condition \(\kappa(G) \geq \alpha(G) – 1\) in [11] to \(\lambda(G) \geq \alpha(G) – 1\) and obtain a similar characterization on non-supereulerian graphs. We also characterize the graph which contains a dominating closed trail under the assumption \(\lambda(G) \geq \alpha'(G) – 2\).

The coloring number \(col(G)\) of a graph \(G\) is the smallest number \(k\) for which there exists a linear ordering of the vertices of \(G\) such that each vertex is preceded by fewer than \(k\) of its neighbors. It is well known that \(\chi(G) \leq col(G)\) for any graph \(G\), where \(\chi(G)\) denotes the chromatic number of \(G\). The Randić index \(R(G)\) of a graph \(G\) is defined as the sum of the weights \(\frac{1}{\sqrt{d(u)d(v)}}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). We show that \(\chi(G) \leq col(G) \leq 2R'(G) \leq R(G)\) for any connected graph \(G\) with at least one edge, and \(col(G) = 2R'(G)\) if and only if \(G\) is a complete graph with some pendent edges attaching to its same vertex, where \(R'(G)\) is a modification of Randić index, defined as the sum of the weights \(\frac{1}{\max\{d(u), d(v)\}}\) of all edges \(uv\) of \(G\). This strengthens a relation between Randić index and chromatic number by Hansen et al. [7], a relation between Randić index and coloring number by Wu et al. [17] and extends a theorem of Deng et al. [2].

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.