Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. Recently, we introduced a new invariant of a graph \(G\), denoted as \(R_5(G)\). Using this invariant and the properties of the adjoint polynomials, we completely determine the adjoint equivalence class of \(\psi_n^3({n-3,1})\). According to the relations between adjoint polynomial and chromatic polynomial, we also simultaneously determine the chromatic equivalence class of \(\psi_n^3({n-3,1})\).

In this article, we prove a conjecture about the equality of two generating functions described in “From Parking Functions to Gelfand Pairs” (Aker, Can, 2012) attached to two sets whose cardinalities are given by Catalan numbers. We establish a combinatorial bijection between the two sets on which the two generating functions were based.

Let \(G\) be a finite cyclic group. Every sequence \(S\) of length \(l\) over \(G\) can be written in the form \(S = (x_1g) + \cdots + (x_lg)\), where \(g \in G\) and \(x_1, \ldots, x_l \in [1, ord(g)]\), and the index \(ind(S)\) of \(S\) is defined to be the minimum of \((x_1 + \cdots + x_l)/ord(g)\) over all possible \(g \in G\) such that \(\langle g \rangle = G\). Recently, the second and third authors determined the index of any minimal zero-sum sequence \(S\) of length \(5\) over a cyclic group of a prime order where \(S =g^2 \cdot (x_2g)\cdot (x_3g)\cdot (x_4g)\). In this paper, we determine the index of any minimal zero-sum sequence \(S\) of length \(5\) over a cyclic group of a prime power order. It is shown that if \(G = \langle g \rangle\) is a cyclic group of prime power order \(n = p^{\mu}\) with \(p \geq 7\) and \(\mu \geq 2\), and \(S = (x_1g) \cdot (x_2g) \cdot (x_3g) \cdot (x_4g) \cdot (x_5g)\) with \(x_1 = x_2\) is a minimal zero-sum sequence with \(\gcd(n, x_1, x_2, x_3, x_4, x_5) = 1\), then \(ind(S) = 2\) if and only if \(S = (mg) \cdot (mg) \cdot (m\frac{n-1}{2}g) \cdot (m\frac{n+3}{2}g) \cdot (m(n-3)g)\) where \(m\) is a positive integer such that \(\gcd(m,n) = 1\).

Let \(G\) be a graph with vertex set \(V(G)\). For any integer \(k \geq 1\), a signed \(k\)-dominating function is a function \(f: V(G) \rightarrow \{-1, 1\}\) satisfying \(\sum_{x \in N[v]} f(t) \geq k\) for every \(v \in V(G)\), where \(N[v]\) is the closed neighborhood of \(v\). The minimum of the values \(\sum_{v \in V(G)} f(v)\), taken over all signed \(k\)-dominating functions \(f\), is called the signed \(k\)-domination number. In this note, we present some new lower bounds on the signed \(k\)-domination number of a graph. Some of our results improve known bounds.

In this paper, we define and study the \(k\)-order Gaussian Fibonacci and Lucas numbers with boundary conditions. We identify and prove the generating functions, the Binet formulas, the summation formulas, matrix representation of \(k\)-order Gaussian Fibonacci numbers, and some significant relationships between \(k\)-order Gaussian Fibonacci and \(k\)-order Lucas numbers, connecting them with usual \(k\)-order Fibonacci numbers.

A vertex-colored path is vertex-rainbow if its internal vertices have distinct colors. For a connected graph \(G\) with connectivity \(\kappa(G)\) and an integer \(k\) with \(1 \leq k \leq \kappa(G)\), the rainbow vertex \(k\)-connectivity of \(G\) is the minimum number of colors required to color the vertices of \(G\) such that any two vertices of \(G\) are connected by \(k\) internally vertex-disjoint vertex-rainbow paths. In this paper, we determine the rainbow vertex \(k\)-connectivities of all small cubic graphs of order \(8\) or less.

For a simple graph \(G = (V, E)\), a vertex labeling \(\alpha: V \rightarrow \{1, 2, \ldots, k\}\) is called a \(k\)-labeling. The weight of an edge \(xy\) in \(G\), denoted by \(w_\phi(xy)\), is the sum of the labels of end vertices \(x\) and \(y\), i.e., \(w_\phi(xy) = \phi(x) + \phi(y)\). A vertex \(k\)-labeling is defined to be an edge irregular \(k\)-labeling of the graph \(G\) if for every two different edges \(e\) and \(f\) there is \(w_\phi(e) \neq w_\phi(f)\). The minimum \(k\) for which the graph \(G\) has an edge irregular \(k\)-labeling is called the edge irregularity strength of \(G\), denoted by \(\mathrm{es}(G)\). In this paper, we determine the exact value for certain families of graphs with path \(P_2\).

We give a \(q\)-analogue of some Dixon-like summation formulas obtained by Gould and Quaintance [Fibonacci Quart. 48 (2010), 56-61] and Chu [Integral Transforms Spec. Funct. 23 (2012), 251-261], respectively. For example, we prove that
\(\sum\limits_{k=0}^{2m} (-1)^{m-k} q^{\binom{m-k}{2}} \binom{2m} {k} \binom{x+k} {2m+r}\binom{x+2m-k} {2m+r}\) = \(\frac{q^{m(x-m-r)}\binom{2m}{m}}{\binom{2m+r}{m}}\binom{x}{m+r}\binom{x+m}{m+r}\) where \(\binom{x}{k}\) denotes the \(q\)-binomial coefficient.

A pentangulation is a simple plane graph such that each face is bounded by a cycle of length \(5\). We consider two diagonal transformations in pentangulations, called \(\mathcal{A}\) and \(\mathcal{B}\). In this paper, we shall prove that any two pentangulations with the same number of vertices can be transformed into each other by \(\mathcal{A}\) and \(\mathcal{B}\). In particular, if they are not isomorphic to a special pentangulation, then we do not need \(\mathcal{B}\).

The harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of the weights of all edges \(uv\) of \(G\), where the weight of \(uv\) is \(\frac{2}{d(u) + d(v)}\), with \(d(u)\) denoting the degree of the vertex \(u\) in \(G\). In this work, we compute the harmonic index of a graph with a cut-vertex and with more than one cut-vertex. As an application, this topological index is computed for Bethe trees and dendrimer trees. Also, the harmonic indices of Fasciagraph and a special type of trees, namely, polytree, are computed.

Let \(G^{\sigma}\) be an oriented graph obtained by assigning an orientation \(\sigma\) to the edge set of a simple undirected graph \(G\). Let \(S(G^{\sigma})\) be the skew adjacency matrix of \(G^{\sigma}\). The skew energy of \(G^{\sigma}\) is defined as the sum of the absolute values of all eigenvalues of \(S(G^{\sigma})\). In this paper, we give the skew energy order of a family of digraphs and determine the oriented bicyclic graphs of order \(n \geq 13\) with the first five largest skew energies, which extends the results of the paper [X. Shen, Y. Hou, C. Zhang, Bicyclic digraphs with extremal skew energy, Electron. J. Linear Algebra 23 (2012) 340-355].

Let \(P_n\) denote the \(n\)-th Catalan-Larcombe-French number. Recently, the \(2\)-log-convexity of the Catalan-Larcombe-French sequence was proved by Sun and Wu. Moreover, they also conjectured that the quotient sequence \(\{\frac{P_{n}}{P_{n-1}}\}_{n= 0}^\infty\) of the Catalan-Larcombe-French sequence is log-concave. In this paper, this conjecture is confirmed by utilizing the upper and lower bounds for \(\frac{P_{n}}{P_{n-1}}\) and finding a middle function \(f(n)\).

It is claimed in [13] that the metric dimension of the Möbius ladder \(M_n\) is \(3\) when \(n \not\equiv 2 \pmod{8}\), but it is wrong; we give a counterexample when \(n \equiv 6 \pmod{8}\). In this paper, we not only give the correct metric dimension in this case but also solve the open problem regarding the metric dimension of \(M_n\) when \(n \equiv 2 \pmod{8}\). Moreover, we conclude that \(M_n\) has two subfamilies with constant metric dimensions.

An edge-colored graph \(G\) is (strong) rainbow connected if any two vertices are connected by a (geodesic) path whose edges have distinct colors. The (strong) rainbow connection number of a connected graph \(G\), denoted by \(\mathrm{src}(G)\) (resp. \(\mathrm{rc}(G)\)), is the smallest number of colors that are needed in order to make \(G\) (strong) rainbow connected. The join \(P_m \vee P_n\) of \(P_m\) and \(P_n\) is the graph consisting of \(P_m\cup P_n\), and all edges between every vertex of \(P_m\) and every vertex of \(P_n\), where \(P_m\) (resp. \(P_n\)) is a path of \(m\) (resp. \(n\)) vertices. In this paper, the precise values of \(\mathrm{rc}(P_m \vee P_n)\) and \(\mathrm{src}(P_m \vee P_n)\) are given for any positive integers \(m\) and \(n\).

Let \(MG(i,n)\) be a connected molecular graph without multiple edges on \(n\)vertices whose minimum degree of vertices is \(i\), where \(i \leq i \leq 4\). One of the newest topological indices is the first Geometric-Arithmetic index. In this paper, we determine the graph with the minimum and the maximum value of the first Geometric-Arithmetic index in the family of graphs \(M{G}(i,n)\),\(l\leq i \leq 3\).

Two graphs are said to be Tutte-equivalent if their Tutte polynomials are equal. In this paper, we provide several different constructions for Tutte-equivalent graphs, including some that are not self-complementary but Tutte-equivalent to their complements (the Akiyama-Harary problem) and some “large” Tutte-equivalent graphs obtained from “small” Tutte-equivalent graphs by \(2\)-sum operations.

Let \(s(n, k) = \binom{6k}{3k} \binom{3k}{k} (\binom{3(n-k)}{n-k} / (2n-1) \binom{3n}{n})\). Recently, Guo confirmed a conjecture of \(Z.-W\). Sun by showing that \(s(n, k)\) is an integer for \(k = 0, 1, \ldots, n\). Let \(d = (3n + 2) / \gcd(3n + 2, 2n – 1)\). In this paper, we prove that \(s(n, k)\) is a multiple of the odd part of \(d\) for \(k = 0, 1, \ldots, n\). Furthermore, if \(\gcd(k, n) = 1\), then \(s(n, k)\) is also a multiple of \(n\). We also show that the \(2\)-adic order of \(s(n, k)\) is at least the sum of the digits in the binary expansion of \(3n\).

For any non-trivial abelian group \(A\) under addition, a graph \(G\) is said to be strong \(A\)-magic if there exists a labeling \(f\) of the edges of \(G\) with non-zero elements of \(A\) such that the vertex labeling \(f^+\) defined as \(f^+(v) = \sum f(uv)\) taken over all edges \(uv\) incident at \(v\) is a constant, and the constant is same for all possible values of \(|V(G)|\). A graph is said to be strong \(A\)-magic if it admits strong \(A\)-magic labeling. In this paper, we consider \((\mathbb{Z}_4, +)\) as an abelian group and we prove strong \(\mathbb{Z}_4\)-magic labeling for various graphs and generalize strong \(\mathbb{Z}_{4p}\)-magic labeling for those graphs. The graphs which admit strong \(\mathbb{Z}_{4p}\)-magic labeling are called as strong \(\mathbb{Z}_{4p}\)-magic graphs.

The well-known Mantel’s Theorem states that a graph on \(n\) vertices and \(m\) edges contains a triangle if \(m > \frac{n^2}{4}\). Nosal proved that every graph on \(m\) edges contains a triangle if the spectral radius \(\lambda_1 > \sqrt{m}\), which is a spectral analog of Mantel’s Theorem. Furthermore, by using Motzkin-Straus Inequality, Nikiforov sharpened Nosal’s result and characterized the extremal graphs when the equality holds. Our first contribution in this note is to give two new proofs of the spectral concise Mantel’s Theorem due to Nikiforov (without help of Motzkin-Straus Inequality). Nikiforov also obtained some results concerning the existence of consecutive cycles and spectral radius. Second, we prove a theorem concerning the existence of consecutive even cycles and spectral radius, which slightly improves a result of Nikiforov. At last, we focus on spectral radius inequalities. Hong proved his famous bound for spectral radius. Later, Hong, Shu, and Fang generalized Hong’s bound to connected graphs with given minimum degree. By using quite different techniques, Nikiforov proved Hong et al.’s bound for general graphs independently. In this note, we prove a new spectral inequality by applying the technique of Nikiforov. Our result extends Stanley’s spectral inequality.

The alliance polynomial of a graph with order \(n\) and maximum degree \(\Delta\) is the polynomial \(A(\Gamma; x) = \sum_{k=-\delta_1}^{\delta_1}A_k(\Gamma) x^{n+k}\), where \(A_k(G)\) is the number of exact defensive \(k\)-alliances in \(G\). We provide an algorithm for computing the alliance polynomial. Furthermore, we obtain some properties of \(A(\Gamma; x)\) and its coefficients. In particular, we prove that the path, cycle, complete, and star graphs are characterized by their alliance polynomials. We also show that the alliance polynomial characterizes many graphs that are not distinguished by other usual polynomials of graphs.

Let \(a\), \(b\), and \(k\) be three nonnegative integers with \(a \geq 2\) and \(b \geq a(k+1)+2\). A graph \(G\) is called a \(k\)-Hamiltonian graph if \(G – U\) contains a Hamiltonian cycle for every subset \(U \subseteq V(G)\) with \(|U| = k\). An \([a, b]\)-factor \(F\) of \(G\) is called a Hamiltonian \([a, b]\)-factor if \(F\) contains a Hamiltonian cycle. If \(G – U\) has a Hamiltonian \([a, b]\)-factor for every subset \(U \subseteq V(G)\) with \(|U| = k\), then we say that \(G\) admits a \(k\)-Hamiltonian \([a, b]\)-factor. Suppose that \(G\) is a \(k\)-Hamiltonian graph of order \(n\) with \(n \geq a+k+2\). In this paper, it is proved that \(G\) includes a \(k\)-Hamiltonian \([a, b]\)-factor if \(\delta(G) \geq a+k\) and \(t(G) \leq a-1+\frac{(a-1)(k+1)}{b-2}\).

Graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks. An embedding \(f\) of a guest graph \(G\) into a host graph \(H\) is a bijection on the vertices such that each edge of \(G\) is mapped into a path of \(H\). In this paper, we introduce a graph called the generalized book and the main results obtained are: (1) For \(r \geq 3\), the minimum wirelength of embedding \(r\)-dimensional hypercube \(Q_r\) into the generalized book \(\mathrm{GB}[2^{r_1}, 2^{r_2}, 2^{r_3}]\), where \(r_1 + r_2 + r_3 = r\). (2) A linear time algorithm to compute the exact wirelength of embedding hypercube into generalized book. (3) An algorithm for embedding hypercube into generalized book with dilation 3, proving that the lower bound obtained by Manuel et al. [28] is sharp.

In this paper, we present a new approach to the convolved Fibonacci numbers arising from the generating function of them and give some new and explicit identities for the convolved Fibonacci numbers.

The generalized Fibonacci cube \(Q_d(f)\) is the graph obtained from the hypercube \(Q_d\) by removing all vertices that contain a given binary word \(f\). A binary word \(f\) is called good if \(Q_d(f)\) is an isometric subgraph of \(Q_d\) for all \(d \geq 1\), and bad otherwise. A non-extendable sequence of contiguous equal digits in a word \(f\) is called a block of \(f\). The question to determine the good (bad) words consisting of at most three blocks was solved by Ilié, Klavžar, and Rho. This question is further studied in the present paper. All the good (bad) words consisting of four blocks are determined completely, and all bad \(2\)-isometric words among consisting of at most four blocks words are found to be \(1100\) and \(0011\).

In this paper, we provide a construction of \(\mathrm{PG}(2,4)\) by a collage of \(\mathrm{AG}(2,3)\) and its dual \(\mathrm{DAG}(2,3)\). Moreover, we prove that the construction is unique.

In this paper, we first present a combinatorial proof of the recurrence relation about the number of the inverse-conjugate compositions of \(2n+1\), \(n > 1\). And then we get some counting results about the inverse-conjugate compositions for special compositions. In particular, we show that the number of the inverse-conjugate compositions of \(4k+1\), \(k > 0\) with odd parts is \(2^k\), and provide an elegant combinatorial proof. Lastly, we give a relation between the number of the inverse-conjugate odd compositions of \(4k+1\) and the number of the self-inverse odd compositions of \(4k+1\).

In this study, by using Jacobsthal and Jacobsthal Lucas matrix sequences, we define \(k\)-Jacobsthal and \(k\)-Jacobsthal Lucas matrix sequences depending on one parameter \(k\). After that, by using two parameters \((s,t)\), we define \((s,t)\)-Jacobsthal and \((s,t)\)-Jacobsthal Lucas matrix sequences. And then, we establish combinatoric representations of all of these matrices.

A graph \(G\) is \(1\)-planar if it can be embedded in the plane \(\mathbb{R}^2\) so that each edge of \(G\) is crossed by at most one other edge. In this paper, we show that each \(1\)-planar graph of maximum degree \(\Delta\) at least \(7\) with neither intersecting triangles nor chordal \(5\)-cycles admits a proper edge coloring with \(\Delta\) colors.

Dirac showed that in a \((k-1)\)-connected graph there is a path through each \(k\) vertices. The path \(k\)-connectivity \(\pi_k(G)\) of a graph \(G\), which is a generalization of Dirac’s notion, was introduced by Hager in 1986. Recently, Mao introduced the concept of path \(k\)-edge-connectivity \(\omega_k(G)\) of a graph \(G\). Denote by \(G \circ H\) the lexicographic product of two graphs \(G\) and \(H\). In this paper, we prove that \(\omega_4(G \circ H) \geq \omega_4(G) |V(H)|\) for any two graphs \(G\) and \(H\). Moreover, the bound is sharp.

A graph \(G = (V(G), E(G))\) is even graceful and equivalently graceful, if there exists an injection \(f\) from the set of vertices \(V(G)\) to \(\{0, 1, 2, 3, 4, \ldots, 2|E(G)|\}\) such that when each edge \(uv\) is assigned the label \(|f(u) – f(v)|\), the resulting edge labels are \(2, 4, 6, \ldots, 2|E(G)|\). In this work, we use even graceful labeling to give a new proof for necessary and sufficient conditions for the gracefulness of the cycle graph. We extend this technique to odd graceful and super Fibonacci graceful labelings of cycle graphs via some number theoretic concept, called a balanced set of natural numbers.

A graph is \(1\)-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. In this paper, we prove that every \(1\)-planar graph without \(4\)-cycles or adjacent \(5\)-vertices is \(5\)-colorable.

In previous researches on classification problems, there are some similar results obtained between \(f\)-coloring and \(g_c\)-coloring. In this article, the author shows that there always are coincident classification results for a regular simple graph \(G\) when the \(f\)-core and the \(g_c\)-core of \(G\) are same and \(f(v) = g(v)\) for each vertex \(v\) in the \(f\)-core (the \(g_c\)-core) of \(G\). However, it is not always coincident for nonregular simple graphs under the same conditions. In addition, the author obtains some new results on the classification problem of \(f\)-colorings for regular graphs. Based on the coincident correlation mentioned above, new results on the classification problem of \(g_c\)-colorings for regular graphs are deduced.

The integrity of a graph \(G = (V, E)\) is defined as \(I(G) = \min\{|S| + m(G-S): S \subseteq V(G)\}\), where \(m(G-S)\) denotes the order of the largest component in the graph \(G-S\). This is a better parameter to measure the stability of a network, as it takes into account both the amount of work done to damage the network and how badly the network is damaged. Computationally, it belongs to the class of intractable problems known as NP-hard. In this paper, we develop a heuristic algorithm to determine the integrity of a graph. Extensive computational experience on \(88\) randomly generated graphs ranging from \(20\%\) to \(90\%\) densities and from \(100\) to \(200\) vertices has shown that the proposed algorithm is very effective.