Let \(G\) be a graph. Let \(g(x)\) and \(f(x)\) be two nonnegative integer-valued functions defined on \(V(G)\) with \(g(x) \leq f(x)\) for any \(x \in V(G)\). A spanning subgraph \(F\) of \(G\) is called a fractional \((g, f)\)-factor if \(g(x) \leq d_G^h(x) \leq f(x)\) 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 \(E_x = \{e : e = xy \in E(G)\}\). A graph \(G\) is said to be fractional \((g, f, n)\)-critical if \(G – N\) has a fractional \((g, f)\)-factor for each \(N \subseteq V(G)\) with \(|N| = n\). In this paper, several sufficient conditions in terms of stability number and degree for graphs to be fractional \((g, f, n)\)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.

The modified Zagreb indices are important topological indices in mathematical chemistry. In this paper, we study the modified Zagreb indices of disjunctions and symmetric differences.

Given a graph \(G\) and a non-negative integer \(g\), the \(g\)-extra-connectivity of \(G\) (written \(\kappa_g(G)\)) is the minimum cardinality of a set of vertices of \(G\), if any, whose deletion disconnects \(G\), and every remaining component has more than \(g\) vertices. The usual connectivity and superconnectivity of \(G\) correspond to \(\kappa_0(G)\) and \(\kappa_1(G)\), respectively. In this paper, we determine \(\kappa_g(P_{n_1} \times P_{n_2} \times \cdots \times P_{n_s})\) for \(0 \leq g \leq s\), where \(\times\) denotes the Cartesian product of graphs. We generalize \(\kappa_g(Q_n)\) for \(0 \leq g \leq n\), \(n \geq 4\), where \(Q_n\) denotes the \(n\)-cube.

A graph labeling is an assignment of integers (labels) to the vertices and/or edges of a graph. Within vertex labelings, two main branches can be distinguished: difference vertex labelings that associate each edge of the graph with the difference of the labels of its endpoints. Graceful and edge-antimagic vertex labelings correspond to these branches, respectively. In this paper, we study some connections between them. Indeed, we study the conditions that allow us to transform any \(a\)-labeling (a special case of graceful labeling) of a tree into an \((a, 1)\)- and \((a, 2)\)-edge antimagic vertex labeling.

The domination number \(\gamma(G)\) of a graph \(G\) is the minimum cardinality among all dominating sets of \(G\), and the independence number \(\alpha(G)\) of \(G\) is the maximum cardinality among all independent sets of \(G\). For any graph \(G\), it is easy to see that \(\gamma(G) \leq \alpha(G)\). In this paper, we present a characterization of trees \(T\) with \(\gamma(T) = \alpha(T)\).

This paper generalizes the concept of locally connected graphs. A graph \(G\) is triangularly connected if for every pair of edges \(e_1, e_2 \in E(G)\), \(G\) has a sequence of \(3\)-cycles \(C_1, C_2, \ldots, C_l\) such that \(e_1 \in C_1, e_2 \in C_l\) and \(E(C_i) \cap E(C_{i+1}) \neq \emptyset\) for \(1 \leq i \leq l-1\). In this paper, we show that every triangularly connected \(K_{1,4}\)-free almost claw-free graph on at least three vertices is fully cycle extendable.

Let \(G = (V,E)\) be a simple graph. \({N}\) and \({Z}\) denote the set of all positive integers and the set of all integers, respectively. The sum graph \(G^+(S)\) of a finite subset \(S \subset{N}\) is the graph \((S, {E})\) with \(uv \in {E}\) if and only if \(u+v \in S\). \(G\) is a sum graph if it is isomorphic to the sum graph of some \(S \subseteq {N}\). The sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices, which result in a sum graph when added to \(G\). By extending \({N}\) to \({Z}\), the notions of the integral sum graph and the integral sum number of \(G\) are obtained, respectively. In this paper, we prove that \(\zeta(\overline{C_n}) = \sigma(\overline{C_n}) = 2n-7\) and that \(\zeta(\overline{W_n}) = \sigma(\overline{W_n}) = 2n-8\) for \(n \geq 7\).

We investigate the relationship between geodetic sets, \(k\)-geodetic sets, dominating sets, and independent sets in arbitrary graphs. As a consequence of the study, we provide several tight bounds on the geodetic number of a graph.

For \(1 \leq d \leq v-1\), let \(V\) denote the \(2v\)-dimensional symplectic space over a finite field \({F}_q\), and fix a \((v-d)\)-dimensional totally isotropic subspace \(W\) of \(V\). Let \({L}(d, 2v) = {P}\cup \{V\}\), where \({P} = \{A \mid A \text{ is a subspace of } V, A \cap W = \{0\} \text{ and } A \subset W^\perp\}\). Partially ordered by ordinary or reverse inclusion, two families of finite atomic lattices are obtained. This article discusses their geometricity, and computes their characteristic polynomials.

Let \(M\) be a graph, and let \(H(M)\) denote the homeomorphism class of \(M\), that is, the set of all graphs obtained from \(M\) by replacing every edge by a `chain’ of edges in series. Given \(M\) it is possible, either using the `chain polynomial’ introduced by E. G. Whitehead and myself (Discrete Math. \(204(1999) 337-356)\) or by ad hoc methods, to obtain an expression which subsumes the chromatic polynomials of all the graphs in \(H(M)\). It is a function of the number of colors and the lengths of the chains replacing the edges of \(M\). This function contains complete information about the chromatic properties of these graphs. In particular, it holds the answer to the question “Which pairs of graphs in \(H(M)\) are chromatically equivalent”. However, extracting this information is not an easy task.

In this paper, I present a method for answering this question. Although at first sight it appears to be wildly impractical, it can be persuaded to yield results for some small graphs. Specific results are given, as well as some general theorems. Among the latter is the theorem that, for any given integer \(\gamma\), almost all cyclically \(3\)-connected graphs with cyclomatic number \(\gamma\) are chromatically unique.

The analogous problem for the Tutte polynomial is also discussed, and some results are given.