By the classical method for obtaining the values of the Riemann zeta-function at even positive integral arguments, we shall give some functional equational proof of some interesting identities and recurrence relations related to the generalized higher-order Euler and Bernoulli numbers attached to a Dirichlet character \(\chi\) with odd conductor \(d\), and shall show an identity between generalized Euler numbers and generalized Bernoulli numbers. Finally, we remark that any weighted short-interval character sums can be expressed as a linear combination of Dirichlet \(L\)-function values at positive integral arguments, via generalized Bernoulli (or Euler) numbers.

A point set \(X\) in the plane is called a k-distance set if there are exactly \(k\) different distances between two distinct points in \(X\). We classify \(11\)-point \(5\)-distance sets.

In this paper, we define the self-inverse sequences related to Sheffer sets and give some interesting results of these sequences. Moreover, we study the self-inverse sequences related to the Laguerre polynomials of order \(a\).

Assume we have a set of \(k\) colors and we assign an arbitrary subset of these colors to each vertex of a graph \(G\). If we require that each vertex to which an empty set is assigned has in its neighborhood all \(k\) colors, then this assignment is called the \(k\)-rainbow dominating function of a graph \(G\). The minimum sum of numbers of assigned colors over all vertices of \(G\), denoted as \(\gamma_{rk}(G)\), is called the \(k\)-rainbow domination number of \(G\). In this paper, we prove that \(\gamma_{r2}(P(n, 3)) \geq \left\lceil \frac{7n}{8} \right\rceil.\)

Let \(G\) be a graph with vertex set \(V(G)\), and let \(k \geq 2\) be an integer. A spanning subgraph \(F\) of \(G\) is called a fractional \(k\)-factor if \(d_G^h(x) = k\) for all \(x \in V(G)\), where \(d_G^h(x) = \sum_{e \in E_x} h(e)\) is the fractional degree of \(x \in V(F)\) with \(E_x = \{e : e = xy, e \in E(G)\}\). The binding number \(bind(G)\) is defined as follows:

\[bind(G) = \min\left\{\frac{|N_G(X)|}{|X|} :\varnothing \neq X \subseteq V(G), N_G(G) \neq V(G)\right\}.\]

In this paper, a binding number condition for a graph to have fractional \(k\)-factors is given.

Let \(\Gamma\) denote a \(d\)-bounded distance-regular graph with diameter \(d \geq 2\). A regular strongly closed subgraph of \(\Gamma\) is said to be a subspace of \(\Gamma\). Define the empty set \(\emptyset\) to be the subspace with diameter \(-1\) in \(\Gamma\). For \(0 \leq i \leq d-1\), let \(\mathcal{L}(\leq i)\) (resp. \(\mathcal{L}(\geq i)\)) denote the set of all subspaces in \(\Gamma\) with diameters \(< i\) (resp. \(\geq i\)) including \(\Gamma\) and \(\emptyset\). If we define the partial order on \(\mathcal{L}(\leq i)\) (resp. \(\mathcal{L}(\geq i)\)) by reverse inclusion (resp. ordinary inclusion), then \(\mathcal{L}(\leq i)\) (resp. \(\mathcal{L}(\geq i)\)) is a poset, denoted by \(\mathcal{L}_R(\leq i)\) (resp. \(\mathcal{L}_o(\geq i)\)). In the present paper, we give the eigenpolynomials of \(\mathcal{L}_R(\leq i)\) and \(\mathcal{L}_o(\geq i)\).

A radio \(k\)-labeling of a connected graph \(G\) is an assignment \(f\) of non-negative integers to the vertices of \(G\) such that

\[|f(x) – f(y)| \geq k + 1 – d(x, y),\]

for any two vertices \(x\) and \(y\), where \(d(x, y)\) is the distance between \(x\) and \(y\) in \(G\). The radio antipodal number is the minimum span of a radio \((diam(G) – 1)\)-labeling of \(G\) and the radio number is the minimum span of a radio \((diam(G))\)-labeling of \(G\).

In this paper, the radio antipodal number and the radio number of the hypercube are determined by using a generalization of binary Gray codes.

In this article, the planes meeting a non-singular quadric of PG\((4,q)\) in a conic are characterized by their intersection properties with points, lines and \(3\)-spaces.

Some Krasnotel’skii-type results previously established for a simply connected orthogonal polygon may be extended to a nonempty compact planar set \(S\) having connected complement. In particular, if every two points of \(S\) are visible via staircase paths from a common point of \(S\), then \(S\) is starshaped via staircase paths. For \(n\) fixed, \(n \geq 1\), if every two points of \(S\) are visible via staircase \(n\)-paths from a common point of \(S\), then \(S\) is starshaped via staircase \((n+1)\)-paths. In each case, the associated staircase kernel is orthogonally convex.

Incorporating the concept of the scattering number and the idea of the vertex-neighbor-connectivity, we introduce a new graph parameter called the vertex-neighbor-scattering number, which measures how easily a graph can be broken into many components with the removal of the neighborhoods of few vertices, and discuss some properties of this parameter. Some tight upper and lower bounds for
this parameter are also given.

This paper is an extension of the work [On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math.
and Comp., \(160 (2005), 125-132.]\), in which for some norms of the circulant matrices with classical Fibonacci and Lucas numbers it is
obtained the lower and upper bounds. In this new paper, we generalize the results of that work.

Let \(a_0, a_1, \ldots, a_{r-1}\) be positive integers and define a conditional sequence \(\{q_n\}\), with initial conditions \(q_0 = 0\) and \(q_1 = 1\), and for all \(n \geq 2\), \(q_n = a_1q_{n-1} + q_{n-2}\) where \(n \equiv t \pmod{r}\). For \(r = 2\), the author studied it in \([1]\). For general \(\{q_n\}\), we found a closed form of the generating function for \(\{q_n\}\) in terms of the continuant in \([2]\). In this paper, we give the matrix representation and a Binet-like formula for the conditional sequence \(\{q_n\}\) by using the matrix methods.

A locally \(nK_2\) graph \(G\) is a graph such that the set of neighbors of any vertex of \(G\) induces a subgraph isomorphic to \(nK_2\). We show that a locally \(nK_2\) graph \(G\) must have at least \(6n – 3\) vertices, and that a locally \(nK_2\) graph with \(6n – 3\) vertices exists if and only if \(n \in \{1, 2, 3, 5\}\), and in these cases the graph is unique up to isomorphism. The case \(n = 5\) is surprisingly connected to a classic theorem of algebraic geometry: The only locally \(5K_2\) graph on \(6 \times 5 – 3 = 27\) vertices is the incidence graph of the 27 straight lines on any nonsingular complex projective cubic surface.

Every graph can be associated to a cheracteristic exponential equation involving powers of (say) \(2\), whose unknowns represent ver-
tex labels and whose general solution is equivalent to a graceful labelling of the graph. If we do not require that the solutions be
integers, we obtain a generalisation of a graceful labelling that uses real numbers as labels. Some graphs that are well known to be non-graceful become graceful in this more general context. Among other things, “real-graceful” labellings provide some information on the Tigidity to be non-graceful, also asymptotically.

Let \(G\) be a graph of order \(n\), and let \(a, b, k\) be nonnegative integers with \(1 \leq a \leq b\). A spanning subgraph \(F\) of \(G\) is called an \([a, b]\)-factor if \(a \leq d_F(x) \leq b\) for each \(x \in V(G)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if \(G – N\) has an \([a, b]\)-factor for each \(N \subseteq V(G)\) with \(|N| = k\). In this paper, it is proved that \(G\) is an \((a, b, k)\)-critical graph if \(n \geq \frac{(a+b-1)(a+b-2)}{b} +\frac{bk}{b-1}\), \(bind(G) \geq \frac{(a+b-1)(n-1)}{b(n-1-k)}\), and \(\delta(G) \neq \left\lfloor \frac{(a-1)n+a+b+bk-2}{a+b-1} \right\rfloor\).

The vulnerability shows the resistance of the network until communication breakdown after the disruption of certain stations or communication links. This study introduces a new vulnerability parameter, neighbor rupture degree. The neighbor rupture degree of a non-complete connected graph \(G\) is defined to be

\[Nr(G) = \max\{w(G/S) – |S| – c(G/S): S \subset V(G), w(G/S) \geq 1\}\]

where \(S\) is any vertex subversion strategy of \(G\), \(w(G/S)\) is the number of connected components in \(G/S\), and \(c(G/S)\) is the maximum order of the components of \(G/S\). In this paper, the neighbor rupture degree of some classes of graphs are obtained and the relations between neighbor rupture degree and other parameters are determined.

A set \(S\) of vertices of a graph \(G = (V, E)\) without isolated vertices is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). The total domination subdivision number \(sd_{\gamma t}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. In this paper, we first prove that \(sd_{\gamma t}(G) \leq n – \delta + 2\) for every simple connected graph \(G\) of order \(n \geq 3\). We also classify all simple connected graphs \(G\) with \(sd_{\gamma t}(G) = n – \delta + 2, n – \delta + 1\), and \(n – \delta\).

In this paper, we obtain the following upper and lower bounds for \(q\)-factorial \([n]_q!\):

\[(q; q)_\infty (1 – q)^{-n} e^{f_q(n+1)} < [n]_q! < (q; q)_\infty (1 – q)^{-n} e^{g_q(n+1)},\] where \(n \geq 1\), \(0 < q < 1\), and the two sequences \(f_q(n)\) and \(g_q(n)\) tend to zero through positive values. Also, we present two examples of the two sequences \(f_q(n)\) and \(g_q(n)\).

A set \(S\) of vertices of a graph \(G = (V, E)\) without isolated vertices is a total dominating set if every vertex of \(V(G)\) is adjacent to some vertex in \(S\). The total domination number \(\gamma_t(G)\) is the minimum cardinality of a total dominating set of \(G\). The total domination subdivision number \(sd_{\gamma t}(G)\) is the minimum number of edges that must be subdivided (each edge in \(G\) can be subdivided at most once) in order to increase the total domination number. In this paper, we first prove that \(sd_{\gamma t}(G) \leq n – \delta + 2\) for every simple connected graph \(G\) of order \(n \geq 3\). We also classify all simple connected graphs \(G\) with \(sd_{\gamma t}(G) = n – \delta + 2, n – \delta + 1\), and \(n – \delta\).

In this paper, we show that the sequences \(p(n, k) := 2^{n-2k} \binom{n-k}{k}\) and \(q(n,k) := 2^{n-2k}\frac{n}{n-k}\binom{n-k}{k}\), \(k = 0, \ldots, \lfloor \frac{n}{2} \rfloor\), are strictly log-concave and then unimodal with at most two consecutive modes. We localize the modes and the integers where there is a plateau. We also give a combinatorial interpretation of \(p(n, k)\) and \(q(n, k)\). These sequences are associated respectively to the Pell numbers and the Pell-Lucas numbers, for which we give some trigonometric relations.

For a finite field \(\mathbb{F}_{p^t}\) of order \(p^t\), where \(p\) is a prime and \(t \geq 1\), we consider the digraph \(G(\mathbb{F}_{p^t}, k)\) that has all the elements of \(\mathbb{F}_{p^t}\) as vertices and a directed edge \(E(a, b)\) if and only if \(a^k = b\), where \(a, b \in \mathbb{F}_{p^t}\). We completely determine the structure of \(G(\mathbb{F}_{p^t},k)\), the isomorphic digraphs of \(\mathbb{F}_{p^t}\), and the longest cycle in \(G(\mathbb{F}_{p^t}, k)\).

Let \(c(H)\) denote the number of components of a graph \(H\). Win proved in \(1989\) that if a connected graph \(G\) satisfies
\[c(G \setminus S) \leq (k – 2)|S| + 2,\text{for every subset S of V(G)},\]
then \(G\) has a spanning tree with maximum degree at most \(k\).

For a spanning tree \(T\) of a connected graph, the \(k\)-excess of a vertex \(v\) is defined to be \(\max\{0, deg_T(v) – k\}\). The total \(k\)-excess \(te(T, k)\) is the summation of the \(k\)-excesses of all vertices, namely,
\[te(T, k) = \sum_{v \in V(T)} \max\{0, deg_T(v) – k\}.\]
This paper gives a sufficient condition for a graph to have a spanning tree with bounded total \(k\)-excess. Our main result is as follows.

Suppose \(k \geq 2\), \(b \geq 0\), and \(G\) is a connected graph satisfying the following condition:
\[\text{for every subset S of V(G)}, \quad c(G \setminus S) \leq (k – 2)|S| + 2+b.\]
Then, \(G\) has a spanning tree with total \(k\)-excess at most \(b\).

A connected graph \(G\) is called \(l_1\)-embeddable, if \(G\) can be isometrically embedded into the \(i\)-space. The hexagonal Möbius graphs \(H_{2m,2k}\) and \(H_{2m+1,2k+1}\) are two classes of hexagonal tilings of a Möbius strip. The regular quadrilateral Möbius graph \(Q_{p,q}\) is a quadrilateral tiling of a Möbius strip. In this note, we show that among these three classes of graphs only \(H_{2,2}\), \(H_{3,3}\), and \(Q_{2,2}\) are \(l_1\)-embeddable.

The boundedness and compactness of the generalized composition operator from \(\mu\)-Bloch spaces to mixed norm spaces are completely characterized in this paper.

By means of inversion techniques, new proofs for Whipple’s transformation and Watson’s \(q\)-Whipple transformation are offered.

In this paper, we introduced the notion of left-right and right-left \(f\)-derivations of a \(B\)-algebra and investigated some related properties. We studied the notion of \(f\)-derivation of a \(0\)-commutative \(B\)-algebra and stated some related properties.

Let \(G\) be a \(k\)-edge connected simple graph with \(k \leq 3\), minimal degree \(\delta(G) \geq 3\), and girth \(g\), where \(r = \left\lfloor \frac{g-1}{2} \right\rfloor\). If the independence number \(\alpha(G)\) of \(G\) satisfies

\[\alpha(G) < \frac{6{(\delta-1)}^{\lfloor\frac{g}{2}\rfloor}-6}{(4-k)(\delta-2)} – \frac{6(g-2r-1)}{4-k} \] then \(G\) is up-embeddable.

Let \(p\) be a prime number such that \(p \equiv 1, 3 \pmod{4}\), let \(\mathbb{F}_p\) be a finite field, and let \(N \in \mathbb{F}_p^* = \mathbb{F}_p – \{0\}\) be a fixed element. Let \(P_p^k(N): x^2 – ky^2 = N\) and \(\tilde{P}_p^k(N): x^2 + 2y – ky^2 = N\) be two Pell equations over \(\mathbb{F}_p\), where \(k = \frac{p-1}{4}\) or \(k = \frac{p-3}{4}\), respectively. Let \(P_p^k(N)(\mathbb{F}_p)\) and \(\tilde{P}_p^k(N)(\mathbb{F}_p)\) denote the set of integer solutions of the Pell equations \(P_p^k(N)\) and \(\tilde{P}_p^k(N)\), respectively. In the first section, we give some preliminaries from the general Pell equation \(x^2 – ky^2 = \pm N\). In the second section, we determine the number of integer solutions of \(P_p^k(N)\). We prove that \(P_p^k(N)(\mathbb{F}_p) = p+1\) if \(p \equiv 1 \pmod{4}\) or \(p \equiv 7 \pmod{12}\) and \(P_p^k(N)(\mathbb{F}_p) = p-1\) if \(p \equiv 11 \pmod{12}\). In the third section, we consider the Pell equation \(\tilde{P}_p^k(N)\). We prove that \(\tilde{P}_p^k(N)(\mathbb{F}_p) = 2p\) if \(p \equiv 1 \pmod{4}\) and \(N \in Q_p\); \(\tilde{P}_p^k(N)(\mathbb{F}_p) = 0\) if \(p \equiv 1 \pmod{4}\) and \(N \notin Q_p\); \(\tilde{P}_p^k(N)(\mathbb{F}_p) = p+1\) if \(p \equiv 3 \pmod{4}\).

For two vertices \(u\) and \(v\) in a strong oriented graph \(D\), the strong distance \(\operatorname{sd}(u,v)\) between \(u\) and \(v\) is the minimum size (the number of arcs) of a strong sub-digraph of \(D\) containing \(u\) and \(v\). For a vertex \(v\) of \(D\), the strong eccentricity \(\operatorname{se}(v)\) is the strong distance between \(v\) and a vertex farthest from \(v\). The strong radius \(\operatorname{srad}(D)\) is the minimum strong eccentricity among the vertices of \(D\). The strong diameter \(\operatorname{sdiam}(D)\) is the maximum strong eccentricity among the vertices of \(D\). In this paper, we investigate the strong distances in strong oriented complete \(k\)-partite graphs. For any integers \(\delta, r, d\) with \(0 \leq \delta \leq \lceil\frac{k}{2}\rceil, 3 \leq r \leq \lfloor\frac{k}{2}\rfloor, 4 \leq d \leq k\), we have shown that there are strong oriented complete \(k\)-partite graphs \(K’, K”, K”’\) such that \(\operatorname{sdiam}(K’) – \operatorname{srad}(K’) = \delta, \operatorname{srad}(K”) = r\), and \(\operatorname{sdiam}(K”’) = d\).

The \(t\)-pebbling number \(f_t(G)\) of a graph \(G\) is the least positive integer \(m\) such that however these \(m\) pebbles are placed on the vertices of \(G\), we can move \(t\) pebbles to any vertex by a sequence of moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. In this paper, we study the generalized Graham’s pebbling conjecture \(f_t(G \times H) \leq f(G)f_t(H)\) for the product of graphs when \(G\) is a complete \(r\)-partite graph and \(H\) has a \(2t\)-pebbling property.

The detour index of a connected graph is defined as the sum of detour distances between all its unordered vertex pairs. We determine the maximum detour index of \(n\)-vertex unicyclic graphs with maximum degree \(\Delta\), and characterize the unique extremal graph, where \(2 \leq \Delta \leq {n-1}\).

In this study, we obtain the relations among \(k\)-Fibonacci, \(k\)-Lucas, and generalized \(k\)-Fibonacci numbers. Then, we define circulant matrices involving \(k\)-Lucas and generalized \(k\)-Fibonacci numbers. Finally, we investigate the upper and lower bounds for the norms of these matrices.

Let \(G = (V(G), E(G))\) be a graph. A set \(S \subseteq V(G)\) is a dominating set if every vertex of \(V(G) – S\) is adjacent to some vertices in \(S\). The domination number \(\gamma(G)\) of \(G\) is the minimum cardinality of a dominating set of \(G\). In this paper, we study the domination number of the circulant graphs \(C(n; \{1, 2\})\), \(C(n; \{1, 3\})\), and \(C(n; \{1, 4\})\) and determine their exact values.

The Merrifield-Simmons index of a graph \(G\), denoted by \(i(G)\), is defined to be the total number of its independent sets, including the empty set. Let \(\theta(a_1, a_2, \ldots, a_k)\) denote the graph obtained by connecting two distinct vertices with \(k\) independent paths of lengths \(a_1, a_2, \ldots, a_k\) respectively, we named it as multi-bridge graphs for convenience. Tight upper and lower bounds for the Merrifield-Simmons index of \(\theta(a_1, a_2, \ldots, a_k)\) are established in this paper.

In this paper, it is shown that the graph \(T_{4}(p, q, r)\) is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum.

In this paper, using the \(q\)-exponential operator technique to two identities due to Jackson, we obtain some \(q\)-series identities involving \(q\)-analogs of \(_{3}{}{\phi}_{2}\).

We consider words \(\pi_1\pi_2\pi_3\ldots\pi_n\) of length \(n\), where \(\pi_i \in \mathbb{N}\) are independently generated with a geometric probability

\[P({\pi} = k) = p(q)^{k-1} \text{where p + q = 1}. \]

Let \(d\) be a fixed non-negative integer. We say that we have an ascent of size \(d\) or more, an ascent of size less than \(d\), a level, and a descent if \({\pi}_{i+1} \geq {\pi}_i+d \), \({\pi}_{i+1} {\pi}_{i+1} \), respectively.We determine the mean and variance of the number of ascents of size less than \(d\) in a random geometrically distributed word. We also show that the distribution is Gaussian as \(n\) tends to infinity.

The graph \(C_n(d; i, j; P_k)\) denotes a cycle \(C_n\) with path \(P_k\) joining two nonconsecutive vertices \(x_i\) and \(x_j\) of the cycle, where \(d\) is the distance between \(x_i\) and \(x_j\) on \(C_n\). In this paper, we obtain that the graph \(C_n(d; i, j; P_k)\) is strongly \(c\)-harmonious when \(k = 2, 3\) and integer \(n \geq 6\).

In this paper, we give several identities of finite sums and some infinite series involving powers and inverse of binomial coefficients.

The concept of integral sum graphs is introduced by Harary \([6]\). A graph \(G\) is an integral sum graph or \(\int\Sigma\)-graph if the vertices of \(G\) can be labelled with distinct integers so that e = uv is an edge of G if and only if the sum of the labels on vertices \(u\) and \(v\) is also a label in G. Xu \([12]\) has shown that the union of any three stars and the union of any number of integral sum trees are integral sum graphs. Xu poses the question as to whether all disconnected forests are integral sum graphs. In this paper, we prove that all banana trees and union of any number of stars are integral sum graphs.

Let \(K_{m} – H\) be the graph obtained from \(K_{m}\) by removing the edges set \(E(H)\) of the graph \(H\) (\(H\) is a subgraph of \(K_{m}\)). We use the symbol \(Z_4\) to denote \(K_4 – P_2\). A sequence \(S\) is potentially \(K_{m} – H\)-graphical if it has a realization containing a \(K_{m} – H\) as a subgraph. Let \(\sigma(K_{m} – H, n)\) denote the smallest degree sum such that every \(n\)-term graphical sequence \(S\) with \(\sigma(S) \geq \sigma(K_{m} – H, n)\) is potentially \(K_{m} – H\)-graphical. In this paper, we determine the values of \(\sigma(K_{r+1} – Z, n)\) for \(n \geq 5r+19, r+1 \geq k \geq 5, j \geq 5\) where \(Z\) is a graph on \(k\) vertices and \(j\) edges which contains a graph \(Z_4\), but not contains a cycle on \(4\) vertices. We also determine the values of \(\sigma(K_{r+1} – Z_4, n)\), \(\sigma(K_{r+1} – (K_4 – e), n)\), \(\sigma(K_{r+1} – K_4, n)\) for \(n \geq 5r+16, r \geq 4\).

A nowhere-zero \(k\)-tension on a graph \(G\) is a mapping from the edges of \(G\) to the set \(\{\pm 1,\pm 2,\ldots,\pm (k-1)\} \subset \mathbb{Z}\) such that, in any fixed orientation of \(G\), for each circuit \(C\) the sum of the labels over the edges of \(C\) oriented in one direction equals the sum of values of the edges of \(C\) oriented oppositely. We show that the existence of an integral tension polynomial that counts nowhere-zero \(k\)-tension on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for tensions on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero tensions on signed graphs with values in an abelian group of odd order. Our results are of two kinds: polynomiality or quasipolynomiality of the tension counting functions, and reciprocity laws that interpret the evaluations of the tension polynomials at negative integers in terms of the combinatorics of the graph.

This paper considered the concepts of monophonic, closed monophonic, and minimal closed monophonic numbers of a connected graph \(G\). It was shown that any positive integers \(m, n, d\), and \(k\) satisfying the conditions that \(4 \leq n \leq m, 3 \leq d \leq k\), and \(k \geq 2m – n + d + 1\) are realizable as the monophonic number, closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Also, any positive integers \(n, m, d\), and \(k\) with \(2 \leq n \leq m, d \geq 3\), and \(k \geq m + d – 1\) are realizable as the closed monophonic number, minimal closed monophonic number, \(m\)-diameter, and order, respectively, of a connected graph. Further, the closed monophonic number of the composition of connected graphs was also determined.

Let \(\Delta(G)\) be the maximum degree of a graph \(G\), and let \(\mathcal{U}(n, \Delta)\) be the set of all unicyclic graphs on \(n\) vertices with fixed maximum degree \(\Delta\). Among all the graphs in \(\mathcal{U}(n, \Delta)\) (\(\Delta \geq \frac{n+3}{2}\)), we characterize the graph with the maximal spectral radius. We also prove that the spectral radius of a unicyclic graph \(G\) on \(n\) (\(n \geq 30\)) vertices strictly increases with its maximum degree when \(\Delta(G) \geq \lceil\frac{7n}{9}\rceil + 1\).

Let \(G\) be a graph, and let \(a\), \(b\) and \(k\) be nonnegative integers with \(1 \leq a \leq b\). An \([a, b]\)-factor of graph \(G\) is defined as a spanning subgraph \(F\) of \(G\) such that \(a \leq d_F(v) \leq b\) for each \(x \in V(G)\). Then a graph \(G\) is called an \((a, b, k)\)-critical graph if after any \(k\) vertices of \(G\) are deleted the remaining subgraph has an \([a, b]\)-factor. In this paper, three sufficient conditions for graphs to be \((a, b, k)\)-critical graphs are given. Furthermore, it is shown that the results in this paper are best possible in some sense.

A total dominating set of a graph \(G\) is a set \(D\) of vertices of \(G\) such that every vertex of \(G\) has a neighbor in \(D\). A vertex of a graph is said to dominate itself and all of its neighbors. A double 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 \(D\). The total (double, respectively) domination number of a graph \(G\) is the minimum cardinality of a total (double, respectively) dominating set of \(G\). We characterize all trees with double domination number equal to total domination number plus one.