Let \(D\) be a digraph with order at least two. The transformation digraph \(D^{++-}\) is the digraph with vertex set \(V(D) \cup A(D)\) in which \((x, y)\) is an arc of \(D^{++-}\) if one of the following conditions holds:(i) \(x, y \in V(D)\), and \((x, y)\) is an arc of \(D\);(ii) \(x, y \in A(D)\), and the head of \(x\) is the tail of \(y\);(iii) \(x \in V(D), y \in A(D)\), and \(x\) is not the tail of \(y\);(iv) \(x \in A(D), y \in V(D)\), and \(y\) is not the head of \(x\).In this paper, we determine the regularity and diameter of \(D^{++-}\). Furthermore, we characterize maximally-arc-connected or super-arc-connected \(D^{++-}\). We also give sufficient conditions for this kind of transformation digraph to be maximally-connected or super-connected.

For a graph \(G\) and any two vertices \(u\) and \(v\) in \(G\), let \(d_G(u,v)\) denote the distance between them and let \(diam(G)\) be the diameter of \(G\). A multi-level distance labeling (or radio labeling) for \(G\) is a function \(f\) that assigns to each vertex of \(G\) a positive integer such that for any two distinct vertices \(u\) and \(v\), \(d_G(u,v) + |f(u) – f(v)| = diam(G) + 1\). The largest positive integer in the range of \(f\) is called the span of \(f\). The radio number of \(G\), denoted \(rn(G)\), is the minimum span of a multi-level distance labeling for \(G\).

A helm graph \(H_n\) is obtained from the wheel \(W_n\) by attaching a vertex of degree one to each of the \(n\) vertices of the cycle of the wheel. In this paper, the radio number of the helm graph is determined for every \(n \geq 3\): \(rn(H_3) = 13\), \(rn(H_4) = 21\), and \(rn(H_n) = 4n + 2\) for every \(n \geq 5\). Also, a lower bound of \(rn(G)\) related to the length of a maximum Hamiltonian path in the graph of distances of \(G\) is proposed.

In this paper, firstly, we define the generalized \(k\)-Horadam sequence and investigate some of its properties. In addition, by also defining the circulant matrix \(C_n(H)\) whose entries are the generalized \(k\)-Horadam numbers, we compute the spectral norm, eigenvalues, and the determinant of this matrix.

The generating function for \(p\)-regular partitions is given by \(\frac{{(q^p;q^p)}_\infty}{{(q;q)}_\infty}\) .In this paper, we will investigate the reciprocal of this generating function. Several interesting results will be presented, and as a corollary of one of these, we will get a parity result due to Sellers for \(p\)-regular partitions with distinct parts.

Motivated by the results from [J. Li, W. Shiu, W. Chan, The Laplacian spectral radius of some graphs, Linear Algebra Appl. \(431 (2009) 99-103]\), we determine the extremal graphs with the second largest Laplacian spectral radius among all bipartite graphs with vertex connectivity \(k\).

Let \(\omega(K_{1,1,t,}{n})\) be the smallest even integer such that every \(n\)-term graphic sequence \(\pi = (d_1,d_2,\ldots,d_n)\) with \(\sigma(\pi) = d_1+d_2+\cdots+d_n \geq \sigma(K_{1,1,t,}{n})\) has a realization \(G\) containing \(K_{1,1,t,}{n}\) as a subgraph, where \(K_{1,1,t,}{n}\) is the \(1 \times 1 \times t\) complete \(3\)-partite graph. Recently, Lai (Discrete Mathematics and Theoretical Computer Science, \(7(2005), 75-81)\) conjectured that for \(n \geq 2t+4\),

\[\sigma(K_{1,1,t,}{n}) = \begin{cases}
(t+1)(n-1)+2 & \text{if \(n\) is odd or \(t\) is odd,}\\
(t+1)(n-1)+1 & \text{if \(n\) and \(t\) are even.}
\end{cases}\]

In this paper, we prove that the above equality holds for \(n \geq t+4\).

A method called the standard construction generates an algebra from a \(K\)-perfect \(m\)-cycle system. Let \({C}_m^K\) denote the class of algebras generated by \(K\)-perfect \(m\)-cycle systems. For each \(m\) and \(K\), there is a known set \(\Sigma_m^K\) of identities which all the algebras in \({C}_m^K\) satisfy. The question of when \({C}_m^K\) is a variety is answered in [2]. When \({C}_m^K\) is a variety, it is defined by \(\Sigma_m^K\). In general, \({C}_m^K\) is a proper subclass of \({V}(\Sigma_m^K)\), the variety of algebras defined by \(\Sigma_m^K\).

If the standard construction is applied to partial \(K\)-perfect \(m\)-cycle systems, then partial algebras result. Using these partial algebras, we are able to investigate properties of \({V}(\Sigma_m^K)\). We show that the free algebras of \({V}(\Sigma_m^K)\) correspond to \(K\)-perfect \(m\)-cycle systems, so \({C}_m^K\) generates \({V}(\Sigma_m^K)\). We also answer two questions asked in [5] concerning subvarieties of \({V}(\Sigma_m^K)\). Many of these results can be unified in the result that for any subset \(K’\) of \(K\), \({V}(\Sigma_m^{K’})\) is generated by the class of algebras corresponding to finite \(K\)-perfect \(m\)-cycle systems.

We examine designs \( \mathcal{D}_i \) and ternary codes \( C_i \), where \( i \in \{112, 113, 162, 163, 274\} \), constructed from a primitive permutation representation of degree 275 of the sporadic simple group \( M^cL \). We prove that \( \dim(C_{113}) = 22, \quad \dim(C_{162}) = 21, \quad C_{113} \supset C_{162}\) and \( M^cL:2 \) acts irreducibly on \( C_{162} \). Furthermore, we have \( C_{112} = C_{163} = C_{274} = V_{27_5}(GF(3)),\) \(
\text{Aut}(\mathcal{D}_{112}) = \text{Aut}(\mathcal{D}_{163})\) = \(
\text{Aut}(\mathcal{D}_{113}) = \text{Aut}(\mathcal{D}_{162}) =
\text{Aut}(C_{113}) = \text{Aut}(C_{162}) = M^{c}L:2 \) while \( Aut(\mathcal{D}_{274}) = Aut(C_{112}) = Aut(C_{163}) = Aut(C_{274}) = S_{275}. \)
We also determine the weight distributions of \( C_{113} \) and \( C_{162} \) and that of their duals.

The purpose of this paper is to investigate some properties of several \(g\)-Bernstein type polynomials to express the bosonic \(p\)-adic \(q\)-integral of those polynomials on \(\mathbb{Z}_p\).

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, it is proved that every \(1\)-planar graph without chordal \(5\)-cycles and with maximum degree \(\Delta \geq 9\) is of class one. Meanwhile, we show that there exist class two \(1\)-planar graphs with maximum degree \(\Delta\) for each \(\Delta \leq 7\).

In \([12]\) Quackenbush has expected that there should be subdirectly irreducible Steiner quasigroups (squags), whose proper homomorphic images are entropic (medial). The smallest interesting cardinality for such squags is \(21\). Using the tripling construction given in \([1]\) we construct all possible nonsimple subdirectly irreducible squags of cardinality \(21\) \((SQ(21)s)\). Consequently, we may say that there are \(4\) distinct classes of nonsimple \(SQ(21)s\), based on the number \(n\) of sub-\(SQ(9)s\) for \(n = 0, 1, 3, 7\). The squags of the first three classes for \(n = 0, 1, 3\) are nonsimple subdirectly irreducible having exactly one proper homomorphic image isomorphic to the entropic \(SQ(3)\) (equivalently, having \(3\) disjoined sub-\(SQ(7)s)\). For \(n = 7\), each squag \(SQ(21\)) of this class has \(3\) disjoint sub-\(SQ(7)s\) and \(7\) sub-\(SQ(9)s\), we will see that this squag is isomorphic to the direct product \(SQ(7)\) \(\times\) \(SQ(3)\). For \(n = 0\), each squag \(SQ(21)\) of this class is a nonsimple subdirectly irreducible having three disjoint sub-\(SQ(7)s\) and no sub-\(SQ(9)s\). In section \(5\), we describe an example for each of these classes. Finally, we review all well-known classes of simple \(SQ(21)s\).

The well-known Petersen graph \(G(5,2)\) admits drawings in the ordinary Euclidean plane in such a way that each edge is represented as a line segment of length \(1\). When two vertices are drawn as the same point in the Euclidean plane, drawings are said to be degenerate. In this paper, we investigate all such degenerate drawings of the Petersen graph and various relationships among them. A heavily degenerate unit distance planar representation, where the representation of a vertex lies in the interior of the representation of an edge it does not belong to, is also shown.

The distance spectral radius of a connected graph \(G\), denoted by \(\rho(G)\), is the maximal eigenvalue of the distance matrix of \(G\). In this paper, we find a sharp lower bound as well as a sharp upper bound of \(\rho(G)\) in terms of \(\omega(G)\), the clique number of \(G\). Furthermore, both extremal graphs are uniquely determined.

Let \(G\) be a graph with \(n\) vertices. The vertex matching polynomial \(M_v(G, x)\) of the graph \(G\) is defined as the sum of \((-1)^rq_v(G,r)x^{n-r}\), in which \(q_v(G,r)\) is the number of \(r\)-vertex independent sets. In this paper, we extend some important properties of the matching polynomial to the vertex matching polynomial \(M_v(G,2x)\). The matching and vertex matching polynomials of some important class of graphs and some applications in nanostructures are presented.

In \([18]\), Farrell and Whitehead investigate circulant graphs that are uniquely characterized by their matching and chromatic polynomials (i.e., graphs that are “matching unique” and “chromatic unique”). They develop a partial classification theorem, by finding all matching unique and chromatic unique circulants on \(n\) vertices, for each \(n \leq 8\). In this paper, we explore circulant graphs that are uniquely characterized by their independence polynomials. We obtain a full classification theorem by proving that a circulant is independence unique if and only if it is the disjoint union of isomorphic complete graphs.

We present a formula for the number of line segments connecting \(q+1\) points of an \(n_1 \times \cdots \times n_k\) rectangular grid. As corollaries, we obtain formulas for the number of lines through at least \(k\) points and, respectively, through exactly \(k\) points of the grid. The well-known case \(k = 2\) is thus generalized. We also present recursive formulas for these numbers assuming \(k = 2, n_1 = n_2\). The well-known case \(q = 2\) is thus generalized.

Let \(H\) and \(G\) be two graphs, where \(G\) is a simple subgraph of \(H\). A \(G\)-decomposition of \(H\), denoted by \((H,G)\)-GD, is a partition of all the edges of \(H\) into subgraphs (\(G\)-blocks), each of which is isomorphic to \(G\). A large set of \((H, G)\)-GD, denoted by \((H,G)\)-LGD, is a partition of all subgraphs isomorphic to \(G\) of \(H\) into \((H,G)\)-GDs. In this paper, we determine the existence spectrums for \((\lambda K_{m,n}, P_3)\)-EGD and \((\lambda K_{n,n,n}, P_3)\)-LGD.

The support of a \(t\)-design is the set of all distinct blocks in the design. The notation \(t-(v,k, \lambda|b^*)\) is used to denote a \(t\)-design with precisely \(b^*\) distinct blocks. We present some results about the structure of support in \(t\)-designs. Some of them are about the number and the range of occurrences of \(i\)-sets (\(1 \leq i \leq t\)) in the support. A new bound for the support sizes of \(t\)-designs is presented. In particular, given a \(t-(v, k, \lambda|b^*)\) design with \(b > b_0\), where \(b\) and \(b_0\) are the cardinality and the minimum cardinality of block sets in the design, respectively, then it is shown that \(b^* \geq \lceil \frac{\lceil \frac{2b}{\lambda}\rceil +7}{2}\rceil\). We also show that when \(\lambda\) varies over all positive integers, then there is no \(t-(v,k,\lambda | b^*)\)-design with the support sizes equal to \(b^*_{min}+1, b^*_{min}+2\) and \(b^*_{min}+3\), where \(b^*_{min}\) denotes the least possible cardinality of the support sizes in this design.

We consider the questions: How many longest cycles must a cubic graph have, and how many may it have? For each \(k \geq 2\) there are infinitely many \(p\) such that there is a cubic graph with \(p\) vertices and precisely one longest cycle of length \(p-k\). On the other hand, if \(G\) is a graph with \(p\) vertices, all of which have odd degree, and its longest cycle has length \(p-1\), then it has a second (but not necessarily a third) longest cycle. We present results and conjectures on the maximum number of cycles in cubic multigraphs of girth \(2, 3, 4\), respectively. For cubic cyclically \(5\)-edge-connected graphs we have no conjecture but, we believe that the generalized Petersen graphs \(P(n, k)\) are relevant. We enumerate the hamiltonian and almost hamiltonian cycles in each \(P(n,2)\). Curiously, there are many of one type if and only if there are few of the other. If \(n\) is odd, then \(P(2n, 2)\) is a covering graph of \(P(n,2)\). (For example, the dodecahedron graph is a covering graph of the Petersen graph). Another curiosity is that one of these has many (respectively few) hamiltonian cycles if and only if the other has few (respectively many) almost hamiltonian cycles.

We study the algebraic properties of soft sets in a hypermodule structure. The concepts of soft hypermodules and soft sub-hypermodules are introduced, and some basic properties are investigated. Furthermore, we define homomorphism and isomorphism of soft hypermodules, and derive three isomorphism theorems of soft hypermodules. By using normal fuzzy sub-hypermodules, three fuzzy isomorphism theorems of soft hypermodules are established.

The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Recently, Heuberger and Wagner [Maximizing the number of independent subsets over trees with bounded degree, J. Graph Theory, \(58 (2008) 49-68\)] investigated the problem of determining the trees with the maximum Merrifield-Simmons index among trees of restricted maximum degree. In this note, we consider the problem of determining the graphs with the maximum Merrifield-Simmons index among connected graphs of restricted minimum degree. Let \(\mathcal{G}_\delta(n)\) denote the set of connected graphs of \(n\) vertices and minimum degree \(\delta\). We first conjecture that among all graphs in \(\mathcal{G}_\delta(n)\), \(n \geq 2\delta\), the graphs with the maximum Merrifield-Simmons index are isomorphic to \(K_{\delta,n-\delta}\) or \(C_5\). Then we affirm this conjecture for the case of \(\delta = 1, 2, 3\).

Cahit and Yilmaz \([15]\) called a graph \(G\) is \(E_k\)-cordial if it is possible to label its edges with numbers from the set \(\{0, 1, \ldots, k-1\}\) in such a way that, at each vertex \(V\) of \(G\), the sum modulo \(k\) of the labels on the edges incident with \(V\) satisfies the inequalities \(|m_(i) – m_(j)| \leq 1\) and \(|n_(i) – n_(j)| \leq 1\), where \(m_(s)\) and \(n_(t)\) are, respectively, the number of edges labeled with \(s\) and the number of vertices labeled with \(t\). In this paper, we give a necessary condition for a graph to be \(E_k\)-cordial for certain \(k\). We also give some new families of \(E_{k}\)-cordial graphs and we prove Lee’s conjecture about the edge-gracefulness of the disjoint union of two cycles.

The Harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of weights \(\frac{2}{d(u)+d(v)}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we consider the Harmonic index of unicyclic graphs with a given order. We give the lower and upper bounds for Harmonic index of unicyclic graphs and characterize the corresponding extremal graphs.

We discuss here some necessary and sufficient conditions for a graph to be prime. We give a procedure to determine whether or not a graph is prime.

The higher order connectivity index is a graph invariant defined as \(^{h}{}{\chi}(G) = \sum_{u_1u_2\ldots u_{h+1}} \frac{1}{\sqrt{{d_{u_1}d_{u_2}\ldots d_{u_{h+1}}}}}\), where the summation is taken over all possible paths of length \(h\) and \(d_{u_i}\) denotes the degree of the vertex \(u_i\) of graph \(G\). In this paper, an exact expression for the fourth order connectivity index of Phenylenes is given.

This paper deals with two types of graph labelings, namely, the super \((a, d)\)-edge antimagic total labeling and super \((a, d)\)-vertex antimagic total labeling on the Harary graph \(C_n^t\). We also construct the super edge-antimagic and super vertex-antimagic total labelings for a disjoint union of \(k\) identical copies of the Harary graph.

The sum-Balaban index of a connected graph \(G\) is defined as

\[J_e(G) = \frac{m}{\mu+1}\sum_{uv \in E(G)} {(D_u + D_v)}^{-\frac{1}{2}},\]

where \(D_u\) is the sum of distances between vertex \(u\) and all other vertices, \(\mu\) is the cyclomatic number, \(E(G)\) is the edge set, and \(m = |E(G)|\). We establish various upper and lower bounds for the sum-Balaban index, and determine the trees with the largest, second-largest, and third-largest as well as the smallest, second-smallest, and third-smallest sum-Balaban indices among the \(n\)-vertex trees for \(n \geq 6\).

A \((v,m,m-1)\)-BIBD \(D\) is said to be near resolvable (NR-BIBD) if the blocks of \(D\) can be partitioned into classes \(R_1, R_2, \ldots, R_v\) such that for each point \(x\) of \(D\), there is precisely one class having no block containing \(x\) and each class contains precisely \(v – 1\) points of the design. If a \((v,m,m-1)\)-NRBIBD has a pair of orthogonal near resolutions, it is said to be doubly resolvable and is denoted DNR\((v,m,m-1)\)-BIBD. A lot of work had been done for the existence of \((v,m,m-1)\)-NRBIBDs, while not so much is known for the existence of DNR\((v,m,m-1)\)-BIBDs except for the existence of DNR\((v,3,2)\)-BIBDs. In this paper, doubly disjoint \((mt+1,m,m-1)\) difference families \(((mt+1,m,m-1)\)-DDDF in short) which were called starters and adders in the previous paper by Vanstone, are used to construct DNR\((v,m,m-1)\)-BIBDs. By using Weil’s theorem on character sum estimates, an explicit lower bound for the existence of a \((mt+1,m,m-1)\)-DDDF and a DNR\((mt+1,m,m-1)\)-BIBD is obtained, where \(mt+1\) is a prime power, \((m,t)=1\). By using this result, it is also proved that there exist a \((v,4,3)\)-DDDF and a DNR\((v,4,3)\)-BIBD for any prime power \(v\equiv 5\pmod{8}\) and \(v\geq 5d\).

For a graph \(G\), Chartrand et al. defined the rainbow connection number \(rc(G)\) and the strong rainbow connection number \(src(G)\) in “G. Chartrand, G.L. John, K.A. McKeon, P. Zhang, Rainbow connection in graphs, Mathematica Bohemica, \(133(1)(2008) 85-98\)”. They raised the following conjecture: for two given positive integers \(a\) and \(b\), there exists a connected graph \(G\) such that \(rc(G) = a\) and \(src(G) = b\) if and only if \(a = b \in \{1,2\}\) or \(3 \leq a \leq b”\). In this short note, we will show that the conjecture is true.

The graph \(P_{a,b}\) is defined as the one obtained by taking \(b\) vertex-disjoint copies of the path \(P_{a+1}\) of length \(a\), coalescing their first vertices into one single vertex labeled \(u\) and then coalescing their last vertices into another single vertex labeled \(v\). K.M. Kathiresan showed that \(P_{2r,2m-1}\) is graceful and conjectured that \(P_{a,b}\) is graceful except when \((a,b) = (2r+1, 4s+2)\). In this paper, an algorithm for finding another graceful labeling of \(P_{2r,2}\) is provided, and \(P_{2r,2(2k+1)}\) is proved to be graceful for all positive \(r\) and \(k\).

A graph \(G\) is \(\)-extendable if every edge is contained in a perfect matching of \(G\). In this note, we prove the following theorem. Let \(d \geq 3\) be an integer, and let \(G\) be a \(d\)-regular graph of order \(n\) without odd components. If \(G\) is not \(1\)-extendable, then \(n \geq 2d + 4\). Examples will show that the given bound is best possible.

A \(k\)-container \(C(u, v)\) of \(G\) between \(u\) and \(v\) is a set of \(k\) internally disjoint paths between \(u\) and \(v\). A \(k\)-container \(C(u,v)\) of \(G\) is a \(k^*\)-container if it contains all nodes of \(G\). A graph \(G\) is \(k^*\)-connected if there exists a \(k^*\)-container between any two distinct nodes. The spanning connectivity of \(G\), \(\kappa^*(G)\), is defined to be the largest integer \(k\) such that \(G\) is \(\omega^*\)-connected for all \(1 \leq \omega \leq k\) if \(G\) is an \(1^*\)-connected graph and undefined if otherwise. A graph \(G\) is super spanning connected if \(\kappa^*(G) = \kappa(G)\). In this paper, we prove that the \(n\)-dimensional augmented cube \(AQ_n\) is super spanning connected.

It is the aim of this paper to explore some new properties of the Padovan sequence using matrix methods. We derive new recurrence relations and generating matrices for the sums of Padovan numbers and \(4n\) subscripted Padovan sequences. Also, we define one type of \((0,1)\) upper Hessenberg matrix whose permanents are Padovan numbers.

In this paper, we prove that every \(n\)-cycle (\(n \geq 6\)) with parallel chords is graceful for all \(n \geq 6\) and every \(n\)-cycle with parallel \(P_k\)-chords of increasing lengths is graceful for \(n \equiv 2 \pmod{4}\) with \(1 \leq k \leq \left\lfloor \frac{n}{2} \right\rfloor – 1\).

On the basis of lit.\([9]\), by the joint tree model, the lower bound of the number of genus embeddings for complete tripartite graph \(K_{n,n,\ell}\) \((\ell \geq m \geq 1)\) is got.

The least common ancestor of two vertices, denoted \(\text{lca}(x, y)\), is a well-defined operation in a directed acyclic graph (dag) \(G\). We introduce \(U_\text{lca}(S)\), a natural extension of \(\text{lca}(x,y)\) for any set \(S\) of vertices. Given such a set \(S_0\), one can iterate \(S_{k+1} = U_\text{lca}(S_k)\) in order to obtain an increasing set sequence. \(G\) being finite, this sequence always has a limit which defines a closure operator. Two equivalent definitions of this operator are given and their relationships with abstract convexity are shown. The good properties of this operator permit to conceive an \(O(n.m)\) time complexity algorithm to calculate its closure. This performance is crucial in applications where dags of thousands of vertices are employed. Two examples are given in the domain of life-science: the first one concerns genes annotations’ understanding by restricting Gene Ontology, the second one deals with identifying taxonomic group of environmental \(DNA\) sequences.

A graph \(G(V,E)\) with order \(p\) and size \(q\) is called \((a,d)\)-edge-antimagic total labeling graph if there exists a bijective function \(f : V(G) \cup E(G) \rightarrow \{1, 2, \ldots, p+q\}\) such that the edge-weights \(\lambda_{f}(uv) = f(u) + f(v) + f(uv)\), \(uv \in E(G)\), form an arithmetic sequence with first term \(a\) and common difference \(d\). Such a labeling is called super if the \(p\) smallest possible labels appear at the vertices. In this paper, we study super \((a, 1)\)-edge-antimagic properties of \(m(P_{4} \square P_{n})\) for \(m, n \geq 1\) and \(m(C_{n} \odot \overline{K_{l}})\) for \(n\) even and \(m, l \geq 1\).

Let \((X, {B})\) be a \(\lambda\)-fold block design with block size \(4\). If a pair of disjoint edges are removed from each block of \(\mathcal{B}\), the resulting collection of \(4\)-cycles \(\mathcal{C}’\) is a partial \(\lambda\)-fold \(4\)-cycle system \((X, \mathcal{C})\). If the deleted edges can be arranged into a collection of \(4\)-cycles \(\mathcal{D}\), then \((X, \mathcal{C} \cup \mathcal{D})\) is a \(\lambda\)-fold \(4\)-cycle system [10]. Now for each block \(b \in {B}\), specify a 1-factorization of \(b\) as \(\{F_1(b), F_2(b), F_3(b)\}\) and define for each \(i = 1, 2, 3\), sets \(\mathcal{C}_i\) and \(\mathcal{D}_i\) as follows: for each \(b \in {B}\), put the \(4\)-cycle \(b \setminus F_i(b)\) in \(\mathcal{C}_i\) and the \(2\) edges belonging to \(F_i(b)\) in \(\mathcal{D}_i\). If the edges in \(\mathcal{D}_i\) can be arranged into a collection of \(4\)-cycles \(\mathcal{D}^*_i\), then \( {M}_i = (X, \mathcal{C}_i \cup \mathcal{D}^*_i)\) is a \(\lambda\)-fold 4-cycle system, called the \(i\)th metamorphosis of \((X, \mathcal{B})\). The full metamorphosis is the set of three metamorphoses \(\{ {M}_1, {M}_2, {M}_3\}\). We give a complete solution of the following problem: for which \(n\) and \(\lambda\) does there exist a \(\lambda\)-fold block design with block size \(4\) having a full metamorphosis into a \(\lambda\)-fold \(4\)-cycle system?

Let \(G\) be a nontrivial connected graph of order \(n\), and \(k\) an integer with \(2 \leq k \leq n\). For a set \(S\) of \(k\) vertices of \(G\), let \(\nu(S)\) denote the maximum number \(\ell\) of edge-disjoint trees \(T_1, T_2, \ldots, T_\ell\) in \(G\) such that \(V(T_i) \cap V(T_j) = S\) for every pair \(i, j\) of distinct integers with \(1 \leq i, j \leq \ell\). Chartrand et al. generalized the concept of connectivity as follows: The \(k\)-connectivity, denoted by \(\kappa_k(G)\), of \(G\) is defined by \(\kappa_k(G) = \min\{\nu(S)\}\), where the minimum is taken over all \(k\)-subsets \(S\) of \(V(G)\). Thus \(\kappa_2(G) = \kappa(G)\), where \(\kappa(G)\) is the connectivity of \(G\). Moreover, \(\kappa_n(G)\) is the maximum number of edge-disjoint spanning trees of \(G\).

This paper mainly focuses on the \(k\)-connectivity of complete bipartite graphs \(K_{a,b}\), where \(1 \leq a \leq b\). First, we obtain the number of edge-disjoint spanning trees of \(K_{a,b}\), which is \(\lfloor \frac{ab}{a+b-1}\rfloor \), and specifically give the \(\lfloor \frac{ab}{a+b-1}\rfloor\) edge-disjoint spanning trees. Then, based on this result, we get the \(k\)-connectivity of \(K_{a,b}\) for all \(2 \leq k \leq a + b\). Namely, if \(k > b – a + 2\) and \(a – b + k\) is odd, then \(\kappa_k(K_{a,b}) =\frac{a+b-k+1}{2} \left\lfloor \frac{(a-b + k + 1)(b-a + k – 1)}{4(k-1)} \right\rfloor\), if \(k > b – a + 2\) and \(a – b + k\) is even, then \(\kappa_k(K_{a,b}) = \frac{a+b-k+1}{2} +\left\lceil \frac{(a – b+ k )(a + b – k)}{4(k-1)} \right\rceil\), and if \(k \leq b – a + 2\), then \(\kappa_k(K_{a,b}) = a\).

A labelling of a graph over a field \(\mathbb{F}\) is a mapping of the edge set of the graph into \(\mathbb{F}\). A labelling is called magic if for any vertex, the sum of the labels of all the edges incident to it is the same. The class of all such labellings forms a vector space over \(\mathbb{F}\) and is called the magic space of the graph. For finite graphs, the dimensional structure of the magic space is well known. In this paper, we give the existence of magic labellings and discuss the dimensional structure of the magic space of locally finite graphs. In particular, for a class of locally finite graphs, we give an explicit basis of the magic space.

For two positive integers \(j\) and \(k\) with \(j \geq k\), an \(L(j,k)\)-labeling of a graph \(G\) is an assignment of nonnegative integers to \(V(G)\) such that the difference between labels of adjacent vertices is at least \(j\), and the difference between labels of vertices that are distance two apart is at least \(k\). The span of an \(L(j, k)\)-labeling of a graph \(G\) is the difference between the maximum and minimum integers used by it. The \(\lambda_{j,k}\)-number of \(G\) is the minimum span over all \(L(j, k)\)-labelings of \(G\). This paper focuses on the \(\lambda_{2,1}\)-number of the Cartesian products of complete graphs. We completely determine the \(\lambda_{2,1}\)-numbers of the Cartesian products of three complete graphs \(K_n\), \(K_m\), and \(K_l\): for any three positive integers \(n\), \(m\), and \(l\).

Let \(G = (V(G), E(G))\) be a graph. A set \(S \subseteq V(G)\) is a packing if for any two vertices \(u\) and \(v\) in \(S\) we have \(d(u, v) \geq 3 \). That is, \(S\) is a packing if and only if for any vertex \(v \in V(G)\), \(|N[v] \cap S| \leq 1\). The packing number \(\rho(G)\) is the maximum cardinality of a packing in \(G\). In this paper, we study the packing number of generalized Petersen graphs \(P(n,2)\) and prove that \(\rho(P(n,2)) = \left\lfloor \frac{n}{7} \right\rfloor + \left\lceil \frac{n+1}{7} \right\rceil + \left\lfloor \frac{n+4}{7} \right\rfloor\) (\(n \geq 5\)).

Let \(G\) be a connected graph. The Wiener index of \(G\) is defined as
\(W(G) = \sum_{u,v \in V(G)} d_G(u,v),\) where \(d_G(u,v)\) is the distance between \(u\) and \(v\) in \(G\) and the summation goes over all the unordered pairs of vertices. In this paper, we investigate the Wiener index of unicyclic graphs with given girth and characterize the extremal graphs with the second maximal and second minimal Wiener index.

This paper uses research methods in the subspace lattices, making a deep research to the lattices of all subsets of a finite set and partition of an n-set. At first, the inclusion relations between different lattices are studied. Then, a characterization of elements contained in a given lattice is given. Finally, the characteristic polynomials of the given lattices are computed.