Let \(G\) be a connected graph on \(n\) vertices. The average eccentricity of a graph \(G\) is defined as \(\varepsilon(G) = \frac{1}{n} \sum_{v \in V(G)} \varepsilon(v)\), where \(\varepsilon(v)\) is the eccentricity of the vertex \(v\), which is the maximum distance from it to any other vertex. In this paper, we characterize the extremal unicyclic graphs among \(n\)-vertex unicyclic graphs having the minimal and the second minimal average eccentricity.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A (defensive) alliance in \(G\) is a subset \(S\) of \(V(G)\) such that for every vertex \(v \in S\), \(|N(v) \cap S| \geq |N(v) \cap (V(G) – S)|\). The alliance partition number of a graph \(G\), \(\psi_a(G)\), is defined to be the maximum number of sets in a partition of \(V(G)\) such that each set is a (defensive) alliance. In this paper, we give both general bounds and exact results for the alliance partition number of graphs, and in particular for regular graphs and trees.

In this paper, we present a unified and simple approach to extremal acyclic graphs without perfect matching for the energy, the Merrifield-Simmons index and Hosoya index.

The notion of equitable coloring was introduced by Meyer in \(1973\). In this paper, we obtain interesting results regarding the equitable chromatic number \(\chi=\) for the sun let graphs \(S_n\), line graph of sun let graphs \(L(S_n)\), middle graph of sun let graphs \(M(S_n)\), and total graph of sun let graphs \(T(S_n)\).

Kühn and Osthus \([2]\) proved that for every positive integer \(\ell\), there exists an integer \(k(\ell) \leq 2^{11}.3\ell^2\), such that the vertex set of every graph \(G\) with \(\delta(G) \geq k(\ell)\) can be partitioned into subsets \(S\) and \(T\) with the properties that \(\delta(G[S]) \geq \ell \leq \delta(G[T])\) and every vertex of \(S\) has at least \(\ell\) neighbors in \(T\). In this note, we improve the upper bound to \(k(\ell) \leq 2^4 – 17\ell^2\).

In this paper, we discuss how the addition of a new edge changes the irregularity strength in \(K(3,n)\), \(tK_3\), and \(tP_4\).

For a graph \(G\), the Merrifield-Simmons index \(i(G)\) and the Hosoya index \(z(G)\) are defined as the total number of independent sets and the total number of matchings of the graph \(G\), respectively. In this paper, we characterize the graphs with the maximal Merrifield-Simmons index and the minimal Hosoya index, respectively, among the bicyclic graphs on \(n\) vertices with a given girth \(g\).

In this paper, we study the existence of \(\alpha\)-labelings for trees by means of particular \((0, 1)\)-matrices called \(a\)-labeling matrices. It is shown that each comet \(S_{k, q}\) admits no \(a\)-labelings whenever \(k > 4(q – 1)\) and \(q \geq 2\). We also give the sufficient conditions for the nonexistence of \(a\)-labelings for trees of diameter at most six. This extends a result of Rosa’s. As a consequence, we prove that \(S_{k, 3}\) has an \(a\)-labeling if and only if \(k \leq 4\).

Given a graph \(G\), an independent set \(I(G)\) is a subset of the vertices of \(G\) such that no two vertices in \(I(G)\) are adjacent. The independence number \(\alpha(G)\) is the order of a largest set of independent vertices. In this paper, we study the independence number for the Generalized Petersen graphs, finding both sharp bounds and exact results for subclasses of the Generalized Petersen graphs.

In this note, we show that the variety of Boolean \(SQS\)-skeins can be defined by a single axiom and, in the process, we find all of the shortest single axioms for said variety. Our investigations were aided by the automated theorem-prover Prover9 and the finite model-finder Mace4.

Let \(G(V,E)\) be a graph. A subset \(S\) of \(V\) is called a dominating set of \(G\) if every vertex in \(V-S\) is adjacent to at least one vertex in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality taken over all dominating sets in \(G\). A dominating set \(S\) of \(G\) is called a complementary perfect dominating set (cpd-set) if the induced subgraph \(\langle V-S \rangle\) has a perfect matching. The complementary perfect domination number, \(\gamma_{cp}(G)\), of \(G\) is the minimum cardinality taken over all cpd-sets in \(G\).

An induced complementary perfect dominating set of a graph (icpd-set) is a dominating set of \(G\) such that the induced subgraph \(\langle V-S \rangle\) has only independent edges. That is, \(\langle V-S \rangle = mK_2\), \(m \geq 1\). The minimum cardinality taken over all such icpd-sets of \(G\) is called the induced complementary perfect domination number of \(G\), and is denoted by \(\gamma_{icp}(G)\).

A subset \(S\) of \(V\) is said to be a complementary connected dominating set (ccd-set) if \(S\) is a dominating set and \(\langle V-S \rangle\) is connected. The complementary connected domination number of a graph is denoted by \(\gamma_{cc}(G)\) and is defined as the minimum number of vertices which form a ccd-set.

It has been proved that \(\gamma_{cp}(G) = n = \gamma_{icp}(G)\) and \(\gamma_{cc}(G) = n-1\) only if \(G\) is a star. And if \(G\) is not a star, then \(\gamma_{cp}, \gamma_{icp}, \gamma_{cc} \leq n-2\). In this paper, we characterize the graphs with \(\gamma_{cc} \leq n-2\), and trees with \(\gamma_{cp} = n-2\) and \(\gamma_{icp} = n-2\).

A graph \(G\) is called \(H\)-equipackable if every maximal \(H\)-packing in \(G\) is also a maximum \(H\)-packing in \(G\). In 2009, \(P_4\)-equipackable paths and cycles, \(M_3\)-equipackable paths and cycles have been characterized. In this paper, \(P_k\)-equipackable paths and cycles, \(M_k\)-equipackable paths and cycles are characterized.

We determine the maximum Wiener index of \(n\)-vertex unicyclic graphs with fixed maximum degree and characterize the unique extremal graph.

The aim of this paper is to define different types of continuities of operators and boundedness of linear operators over fuzzy \(n\)-normed linear spaces. Also, some definitions such as fuzzy continuity, sequential fuzzy continuity, weakly fuzzy continuity, strongly fuzzy continuity, weakly fuzzy boundedness, and strongly fuzzy boundedness are given in fuzzy \(n\)-normed linear spaces. In addition, some theorems related to these definitions are proved.

In this paper, we study the enumeration of noncrossing partitions with fixed points. The expressions of \({f_m}(x_1, x_2,x_3, 0, \ldots, 0)\) and \({f_m}(x_1, x_2, 0, \ldots, 0, x_{p+3}, 0, \ldots, 0)\) are found, and a new proof of the expression of \({f_m}(x_1, x_2,0, 0, \ldots, 0)\) is obtained using diophantine equations.

Let \(G\) be a subgraph of \(K_n\). The graph obtained from \(G\) by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a \(T(G)\)-triple. An edge-disjoint decomposition of \(3K_n\) into copies of \(T(G)\) is called a \(T(G)\)-triple system of order \(n\). If, in each copy of \(T(G)\) in a \(T(G)\)-triple system, one edge is taken from each 3-cycle (chosen so that these edges form a copy of \(G\)) in such a way that the resulting copies of \(G\) form an edge-disjoint decomposition of \(K_n\), then the \(T(G)\)-triple system is said to be perfect. The set of positive integers \(n\) for which a perfect \(T(G)\)-triple system exists is called its spectrum. Earlier papers by authors including Billington, Lindner, Kıvcıkgızı, and Rosa determined the spectra for cases where \(G\) is any subgraph of \(K_4\). In this paper, we will focus on the star graph \(K_{1,k}\) and discuss the existence of perfect \(T(K_{1,k})\)-triple systems. Especially, for prime powers \(k\), its spectra are completely determined.

In this paper, we investigate some basic properties of these eight kinds of transformation digraphs.

For any given \(k\)-uniform list assignment \(L\), a graph \(G\) is equitably \(k\)-choosable if and only if \(G\) is \(\ell\)-colorable and each color appears on at most \(\lceil \frac{|V(G)|}{k} \rceil\) vertices. A graph \(G\) is equitably \(\ell\)-colorable if \(G\) has a proper vertex coloring with \(k\) colors such that the size of the color classes differ by at most \(1\). In this paper, we prove that every planar graph \(G\) without \(6\)- and \(7\)-cycles is equitably \(k\)-colorable and equitably \(k\)-choosable where \(k \geq \max\{\Delta(G), 6\}\).

This paper introduces the concepts of forcing \(m\)-convexity number and forcing clique number of a graph. We show that the forcing \(m\)-convexity numbers of some Cartesian product and composition of graphs are related to the forcing clique numbers of the graphs. We also show that the forcing \(m\)-convexity number of the composition \(G[K_n]\), where \(G\) is a connected graph with no extreme vertex, is equal to the forcing \(m\)-convexity number of \(G\).

A spectrally arbitrary pattern \({A}\) is a sign pattern of order \(n\) such that every monic real polynomial of degree \(n\) can be achieved as the characteristic polynomial of a matrix with sign pattern \({A}\). A sign pattern \({A}\) is minimally spectrally arbitrary if it is spectrally arbitrary but is not spectrally arbitrary if any nonzero entry (or entries) of \({A}\) is replaced by zero. In this paper, we introduce some new sign patterns which are minimally spectrally arbitrary for all orders \(n\geq 7\).

Let \(G\) be a graph with vertex-set \(V = V(G)\) and edge-set \(E = E(G)\), and let \(e = |E(G)|\) and \(v = |V(G)|\). A one-to-one map \(\lambda\) from \(V \cup E\) onto the integers \(\{1, 2, \ldots, v+e\}\) is called a vertex-magic total labeling if there is a constant \(k\) so that for every vertex \(x\),

\[\lambda(x) + \sum \lambda(xy) = k\]

where the sum is over all edges \(xy\) where \(y\) is adjacent to \(x\). Let us call the sum of labels at vertex \(x\) the weight \(w_\lambda\) of the vertex under labeling \(\lambda\); we require \(w_\lambda(x) = k\) for all \(x\). The constant \(k\) is called the magic constant for \(\lambda\).

A sun \(S_n\) is a cycle on \(n\) vertices \(C_n\), for \(n \geq 3\), with an edge terminating in a vertex of degree \(1\) attached to each vertex.

In this paper, we present the vertex-magic total labeling of the union of suns, including the union of $m$ non-isomorphic suns for any positive integer $m \geq 3$, proving the conjecture given in [6].

The Randić index of an organic molecule whose molecular graph is \(G\) is the sum of the weights \((d(u)d(v))^{1/2}\) of all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of the vertex \(u\) of the molecular graph \(G\). Among all trees with \(n\) vertices and \(k\) pendant vertices, the extremal trees with the minimum, the second minimum, and the third minimum Randić index were characterized by Hansen, Li, and Wu \(et al\)., respectively. In this paper, we further investigate some small Randić index properties and give other elements of small Randić index ordering of trees with \(k\) pendant vertices.

Consider a complete graph of multiplicity \(2\), where between every pair of vertices there is one red and one blue edge. Can the edge set of such a graph be decomposed into isomorphic copies of a \(2\)-coloured path of length \(2k\) that contains \(k\) red and\(k\) blue edges? A necessary condition for this to be true is \(n(n-1) \equiv 0 \mod k\). We show that this is sufficient for \(k \leqq 3\).

In this paper, we investigate super-simple cyclic \((v, k, \lambda)\)-BIBDs (SCBIBs). Some general constructions for SCBIBs are given. The spectrum of super-simple cyclic \((v, 3, \lambda)\) is completely determined for \(\lambda = 2, 3\) and \(v – 2\). From that, some new optical orthogonal codes are obtained.

The cycle structure of a Latin square autotopism \(\Theta = (\alpha, \beta, \gamma)\) is the triple \((I_\alpha,I_\beta, I_\gamma)\), where \(I_\delta\) is the cycle structure of \(\delta\), for all \(\delta \in \{\alpha, \beta, \gamma\}\). In this paper, we study some properties of these cycle structures and, as a consequence, we give a classification of all autotopisms of the Latin squares of order up to \(11\).

This work presents explicit expressions of the \(3\)-restricted edge connectivity of Cartesian product graphs, which yields some sufficient conditions for the product graphs to be maximally \(3\)-restricted edge connected.

Dirac characterized chordal graphs by every minimal \((2\)-)vertex separator inducing a complete subgraph. This generalizes to \(k\)-vertex separators and to a characterization of the class of \(\{P_5, 2P_3\}\)-free chordal graphs. The correspondence between minimal \(2\)-vertex separators of chordal graphs and the edges of their clique trees parallels a correspondence between minimal \(k\)-vertex separators of \(\{P_5, 2P_3\}\)-free chordal graphs and certain \((k-1)\)-edge substars of their clique trees.

It is well known that the Petersen graph does not contain a Hamilton cycle. In \(1983\), Alspach completely determined which Generalized Petersen graphs are Hamiltonian \([1]\). In this paper, we define a larger class of graphs which includes the Generalized Petersen graphs as a special case, and determine which graphs in this larger class are Hamiltonian, and which are \(1\)-factorable. We call this larger class spoked Cayley graphs.

Let \(K_v\) be the complete graph with \(v\) vertices, where any two distinct vertices \(x\) and \(y\) are joined by exactly one edge \(\{x,y\}\). Let \(G\) be a finite simple graph. A \(G\)-design of \(K_v\), denoted by \((v,G,1)\)-GD, is a pair \((X,\mathcal{B})\), where \(X\) is the vertex set of \(K_v\), and \(\mathcal{B}\) is a collection of subgraphs of \(K_v\), called blocks, such that each block is isomorphic to \(G\) and any two distinct vertices in \(K_v\) are joined in exactly one block of \(\mathcal{B}\). In this paper, the discussed graphs are \(G_i\), \(i = 1,2,3,4\), where \(G_i\) are the four graphs with 7 points, 7 edges, and a 5-cycle. We obtain the existence spectrum of \((v, G_i,1)\)-GD.

Let \(\text{ASG}(2v,\mathbb{F}_q)\) be the \(2v\)-dimensional affine-symplectic space over the finite field \(\mathbb{F}_q\), and let \(\text{ASp}_{2v}(\mathbb{F}_q)\) be the affine-symplectic group of degree \(2v\) over \(\mathbb{F}_q\). For any two orbits \(M’\) and \(M”\) of flats under \(\text{ASp}_{2v}(\mathbb{F}_q)\), let \(\mathcal{L}’\) (resp. \(\mathcal{L}”\)) be the set of all flats which are joins (resp. intersections) of flats in \(M’\) (resp. \(M”\)) such that \(M” \subseteq L’\) (resp. \(M’ \subseteq \mathcal{L}”\)) and assume the join (resp. intersection) of the empty set of flats in \(\text{ASG}(2v,\mathbb{F}_q)\) is \(\emptyset\) (resp. \(\mathbb{F}_q^{(2v)}\)). Let \(\mathcal{L} =\mathcal{L}’ \cap \mathcal{L}”\). By ordering \(\mathcal{L}’,\mathcal{L}”, \mathcal{L}\) by ordinary or reverse inclusion, six lattices are obtained. This article discusses the relations between different lattices, and computes their characteristic polynomial.

In this paper, we calculate the number of fuzzy subgroups of a special class of non-abelian groups of order \(p^3\).

This paper addresses the problem of capturing nondominated points on non-convex Pareto frontiers, which are encountered in \(E\)-convex multi-objective optimization problems. We define a nondecreasing map \(T\) which transfers a non-convex Pareto frontier to a convex Pareto frontier. An algorithm to find a piecewise linear approximation of the nondominated set of the convex Pareto frontier is applied. Finally, the inverse map of \(T\) is used to obtain the non-convex Pareto frontier.

The aim of our paper is to introduce generalized neighborhood bases and \(gn-T_2\)-spaces. \((\psi, \psi’)\)-continuity, sequentially \((\psi, \psi’)\)-continuity, and \(\psi\)-convergency are investigated on strong generalized first countable spaces, and also two results about \(\psi\)-convergency on \((\psi, \psi’)\)-\(T_2\)-spaces are given.

For a graph \(H\) and an integer \(k \geq 2\), let \(\sigma_k(H)\) denote the minimum degree sum of \(k\) independent vertices of \(H\). We prove that if a connected claw-free graph \(G\) satisfies \(\sigma_{k+1}(G) \geq |G| – k\), then \(G\) has a spanning tree with at most \(k\) leaves. We also show that the bound \(|G| – k\) is sharp and discuss the maximum degree of the required spanning trees.

Define the conditional recurrence sequence \(q_n = aq_{n-1} + bq_{n-2}\) if \(n\) is even, \(q_n = bq_{n-1} + cq_{n-2}\) if \(n\) is odd, where \(q_0 = 0, q_1 = 1\). Then \(q_n\) satisfies a fourth-order recurrence while both \(q_{2n}\) and \(q_{2n+1}\) satisfy a second-order recurrence.

Analogously to a Lucas pseudoprime, we define a composite number \(n\) to be a conditional Lucas pseudoprime (clpsp) if \(n\) divides \(q_{n – (\frac{\Delta}{n})}\), where \(\Delta = a^2 + b^2 + 4ab\) and \((\frac{\Delta}{n})\) denotes the Jacobi symbol. We prove that if \((n, 2ab\Delta) = 1\), then there are infinitely many conditional Lucas pseudoprimes. We also address the question, given an odd composite integer \(n\), for how many pairs \((a, b)\) is \(n\) a conditional Lucas pseudoprime?

Let \(G\) be a simple connected graph with \(n\) vertices. Denoted by \(L(G)\) the Laplacian matrix of G. In this paper, we present a sequence of graphs \({G_n}\) with \(\lim\limits_{n\to \infty} \mu_3(G_n) = 1.5550\) by investigating the eigenvalues of the line graphs of \({G_n}\). Moreover, we prove that the limit is the minimal limit point of the third largest Laplacian eigenvalues of graphs.

Two cycles are said to be intersecting if they share at least one common vertex. Let \(\chi'(G)\) and \(\chi”(G)\) denote the list edge chromatic number and list total chromatic number of a graph \(G\), respectively.In this paper, we proved that for any toroidal graph G without intersecting triangles, \(\chi'(G) \leq \Delta(G) +1\) and \(\chi”(G) \leq \Delta(G)+2\) if \(\Delta(G) \geq 6\), and \(\chi'(G) = \Delta(G)\) if \(\Delta(G) \geq 8\).

Graphs which are derived from the same graph are called homeomorphic graphs or simply homeomorphs. A \(K_4\)-homeomorph denoted by
\(K_4(a,,c,d,e, f)\) is obtained by subdividing the six paths of a complete graph with four vertices into \(a, b, c,d, e, f\) number of segments, respectively.In this paper, we shall study the chromaticity of \(K_4(a, b,c,d,e, f)\) with exactly two non-adjacent paths of length two. We also give a sufficient and necessary condition for all the graphs in this family to be chromatically
unique.

Let G be a graph with diameter d. An antipodal labeling of G is a function f that assigns to each vertex a
non-negative integer (label) such that for any two vertices \(u\) and \(v\), \(|f(u) — f(v)| \geq d — d(u,v)\), where \(d(u, v)\)
is the distance between \(u\) and \(v\). The span of an antipodal labeling f is \(\max{f(u) — f(v) : u,v \in V(G)}\). The
antipodal number for G, denoted by an\((G)\), is the minimum span of an antipodal labeling for \(G\). Let \(C_n\) denote
the cycle on n vertices. Chartrand \(et al\). \([4]\) determined the value of an\((C_n)\) for \(n \equiv 2 \pmod 4\). In this article we
obtain the value of an\((C_n)\) for \(n \equiv 1 \pmod 4\), confirming a conjecture in \([4]\). Moreover, we settle the case \(n \equiv 3 \pmod 4\), and improve the known lower bound and give an upper bound for the case \(n \equiv 0 \pmod 4\).

We classify all embeddings \(\theta\) : \(PG(n,\mathbb{K}) \rightarrow PG(d,\mathbb{F})\), with \(d \geq \frac{n(n+1)}{2}\)
and \(\mathbb{K},\mathbb{F}\) skew fields with \(|\mathbb{K}| > 2\), such that \(\theta\) maps the set of points of each line of \(PG(n, \mathbb{K})\) to a set of coplanar points of \(PG(n, \mathbb{F})\), and such that the image of \(\theta\) generates \(PG(d, \mathbb{F})\). It turns out that \(d = \frac{1}{2}n(n + 3)\) and all examples “essentially” arise from a similar “full” embedding \(\theta’\) : \(PG(n, \mathbb{K}) \rightarrow PG(d, \mathbb{K})\) by identifying \(\mathbb{K}\) with subfields of F and embedding \(PG(d, \mathbb{K})\) into \(PG(d, \mathbb{F})\) by several ordinary field extensions. These “full” embeddings satisfy one more property and are classified in \([5]\). They relate to the quadric Verone-sean of \(PG(n, \mathbb{K})\) in \(PG(d, \mathbb{K})\) and its projections from subspaces of \(PG(n, \mathbb{K})\) generated by sub-Veroneseans (the point sets corresponding to subspaces of \(PG(n, \mathbb{K})\), if \(\mathbb{K}\) is commutative, and to a degenerate analogue of this, if \(\mathbb{K}\) is noncommutative.

Let \(\mathbb{N}\) be the set of all positive integers, and \(\mathbb{Z}_n = \{0, 1, 2, \ldots, n-1\}\). For any \(h \in \mathbb{N}\), a graph \(G = (V, E)\) is said to be \(\mathbb{Z}_h\)-magic if there exists a labeling \(f: E \rightarrow \mathbb{Z}_h \setminus \{0\}\) such that the induced vertex labeling \(f^+: V \rightarrow \mathbb{Z}_h\), defined by \(f^+(v) = \sum_{uv \in E(v)} f(uv)\), is a constant map. The integer-magic spectrum of \(G\) is the set \(\text{JM}(G) = \{h \in \mathbb{N} \mid G \text{ is } \mathbb{Z}_h\text{-magic}\}\). A sun graph is obtained from attaching a path to each pair of adjacent vertices in an \(n\)-cycle. In this paper, we show that the integer-magic spectra of sun graphs are completely determined.

Let \(e: \mathcal{S} \rightarrow \Sigma\) be a full polarized projective embedding of a dense near polygon \(\mathcal{S}\), i.e., for every point \(p\) of \(\mathcal{S}\), the set \(H_p\) of points at non-maximal distance from \(p\) is mapped by \(e\) into a hyperplane \(\Pi_p\) of \(\Sigma\). We show that if every line of \(S\) is incident with precisely three points or if \(\mathcal{S}\) satisfies a certain property (P\(_y\)) then the map \(p \mapsto \Pi_p\) defines a full polarized embedding \(e^*\) (the so-called dual embedding of \(e\)) of \(\mathcal{S}\) into a subspace of the dual \(\Sigma^*\) of \(\Sigma\). This generalizes a result of \([6]\) where it was shown that every embedding of a thick dual polar space has a dual embedding. We determine which known dense near polygons satisfy property (P\(_y\)). This allows us to conclude that every full polarized embedding of a known dense near polygon has a dual embedding.

Let \(\mathcal{B}(n,k)\) be the set of bicyclic graphs with \(n\) vertices and \(k\) pendant vertices. In this paper, we determine the unique graph with minimal least eigenvalue among all graphs in \(\mathcal{B}(n,k)\). This extremal graph is the same as that on the Laplacian spectral radius as done by Ji-Ming Guo(The Laplacian spectral radius of bicyclic graphsmwith \(n\) vertices and \(k\) pendant vertices, Science China Mathematics, \(53(8)(2010)2135-2142]\). Moreover, the minimal least eigenvalue is a decreasing function on \(k\).

Gnanajothi conjectured that all trees are odd-graceful and verified this conjecture for all trees with order up to \(10\). Since the
conjecture is open now we present a proof to the odd-gracefulness of all lobsters and show a connection between set-ordered odd-graceful labellings and bipartite graceful labellings in a connected graph.

In this article, the lines not meeting a hyperbolic quadric in PG\((3,q)\) are characterized by their intersection properties with points and planes.