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.

Let \(G\) be a simple graph of order \(p \geq 2\). A proper \(k\)-total coloring of a simple graph \(G\) is called a \(k\)-vertex distinguishing proper total coloring (\(k\)-VDTC) if for any two distinct vertices \(u\) and \(v\) of \(G\), the set of colors assigned to \(u\) and its incident edges differs from the set of colors assigned to \(v\) and its incident edges. The notation \(\chi_{vt}(G)\) indicates the smallest number of colors required for which \(G\) admits a \(k\)-VDTC with \(k \geq \chi_{vt}(G)\). For every integer \(m \geq 3\), we will present a graph \(G\) of maximum degree \(m\) such that \(\chi_{vt}(G) < \chi_{vt}(H)\) for some proper subgraph \(H \subseteq G\).

Let \(G = (V,E)\) be a graph. Let \(\gamma(G)\) and \(\gamma_t(G)\) be the domination and total domination number of a graph \(G\), respectively. The \(\gamma\)-criticality and \(\gamma_t\)-criticality of Harary graphs are studied. The Question \(2\) of the paper [W. Goddard et al., The Diameter of total domination vertex critical graphs, Discrete Math. \(286 (2004), 255-261]\) is fully answered with the family of Harary graphs. It is answered to the second part of Question \(1\) of that paper with some Harary graphs.

Let \(G\) be a connected graph. The hyper-Wiener index \(WW(G)\) is defined as \(WW(G) = \frac{1}{2}\sum_{u,v \in V(G)} d(u,v) + \frac{1}{2} \sum_{u,v \in V(G)} d^2(u,v),\) with the summation going over all pairs of vertices in \(G\) and \(d(u,v)\) denotes the distance between \(u\) and \(v\) in \(G\). In this paper, we determine the upper or lower bounds on hyper-Wiener index of trees with given number of pendent vertices, matching number, independence number, domination number, diameter, radius, and maximum degree.

A large set of resolvable Mendelsohn triple systems of order \(v\), denoted by \(\text{LRMTS}(v)\), is a collection of \(v-2\) \(\text{RMTS}(v)\)s based on \(v\)-set \(X\), such that every Mendelsohn triple of \(X\) occurs as a block in exactly one of the \(v-2\) \(\text{RMTS}(v)\)s. In this paper, we use \(\text{TRIQ}\) and \(\text{LR-design}\) to present a new product construction for \(\text{LRMTS}(v)\)s. This provides some new infinite families of \(\text{LRMTS}(v)\)s.

In this paper, we investigate the existence of nontrivial solutions for the equation \(y(G \Box H) – \gamma(G) \gamma(H)\) fixing one factor. For the complete bipartite graphs \(K_{m,n}\), we characterize all nontrivial solutions when \(m = 2, n \geq 3\) and prove the nonexistence of solutions when \(m \geq 2, n \leq 3\). In addition, it is proved that the above equation has no nontrivial solution if \(A\) is one of the graphs obtained from \(G\), the cycle of length \(n\), either by adding a vertex and one pendant edge joining this vertex to any vertex to any \(v\in V(C_n)\), or by adding one chord joining two alternating vertices of \(C_n\).

For a graph \(G = (V(G), E(G))\), let \(i(G)\) be the number of isolated vertices in \(G\). The isolated toughness of \(G\) is defined as
\(I(G) = \min\left\{\frac{|S|}{i(G-S)}: S \subseteq V(G), i(G-S) \geq 2\right\}\) if \(G\) is not complete; \(I(G) = |V(G)|-1\) otherwise. In this paper, several sufficient conditions in terms of isolated toughness are obtained for the existence of \([a, b]\)-factors avoiding given subgraphs, e.g., a set of vertices, a set of edges and a matching, respectively.

In a graph \(G\), the distance \(d(u,v)\) between a pair of vertices \(u\) and \(v\) is the length of a shortest path joining them. The eccentricity \(e(u)\) of a vertex \(u\) is the distance to a vertex farthest from \(u\). The minimum eccentricity is called the radius of the graph and the maximum eccentricity is called the diameter of the graph. The radial graph \(R(G)\) based on \(G\) has the vertex set as in \(G\). Two vertices \(u\) and \(v\) are adjacent in \(R(G)\) if the distance between them in \(G\) is equal to the radius of \(G\). If \(G\) is disconnected, then two vertices are adjacent in \(R(G)\) if they belong to different components. The main objective of this paper is to find a necessary and sufficient condition for a graph to be a radial graph.

Let \(\{T, T’\}\) be a Latin bitrade. Then \(T\) (and \(T’\)) is said to be \((r,c,e)\)-homogeneous if each row contains precisely \(r\) entries, each column contains precisely \(c\) entries, and each entry occurs precisely \(e\) times. An \((r,c,e)\)-homogeneous Latin bitrade can be embedded on the torus only for three parameter sets, namely \((r,c,e) = (3,3,3), (4,4,2)\), or \((6,3,2)\). The first case has been completely classified by a number of authors. We present classifications for the other two cases.

In this paper, we prove an interesting property of rook polynomials for \(2\)-D square boards and extend that for rook polynomials for \(3\)-D cubic, and \(r\)-D “hypercubic” boards. In particular, we prove that for \(r\)-D rook polynomials the modulus of the sum of their roots equals their degree. We end with some further questions, mainly for the \(2\)-D and \(3\)-D case, that could serve as future projects.

Let \(G\) be a finite graph and \(H\) be a subgraph of \(G\). If \(V(H) = V(G)\), then the subgraph \(H\) is called a \({spanning \;subgraph}\) of \(G\). A spanning subgraph \(H\) of \(G\) is called an \({F-factor}\) if each component of \(H\) is isomorphic to \(F\). Further, if there exists a subgraph of \(G\) whose vertex set is \(\lambda V(G)\) and can be partitioned into \(F\)-factors, then it is called a \({\lambda-fold \;F-factor}\) of \(G\), denoted by \(S_\lambda(1,F,G)\). A \({large \; set}\) of \(\lambda\)-fold \(F\)-factors in \(G\) is a partition \(\{\mathcal{B}_i\}_{i}\) of all subgraphs of \(G\) isomorphic to \(F\), such that each \((X, \mathcal{B}_i)\) forms a \(\lambda\)-fold \(F\)-factor of \(G\). In this paper, we investigate the large set of \(\lambda\)-fold \(P_3\)-factors in \(K_{v,v}\) and obtain its existence spectrum.

Let \(k \geq 1\), \(l \geq 3\), and \(s \geq 5\) be integers. In \(1990\), Erdős and Faudree conjectured that if \(G\) is a graph of order \(4k\) with \(\delta(G) \geq 2k\), then \(G\) contains \(k\) vertex-disjoint \(4\)-cycles. In this paper, we consider an analogous question for \(5\)-cycles; that is to say, if \(G\) is a graph of order \(5k\) with \(\delta(G) \geq 3k\), then \(G\) contains \(k\) vertex-disjoint \(5\)-cycles? In support of this question, we prove that if \(G\) is a graph of order \(5k\) with \(\omega_2(G) \geq 6l – 2\), then, unless \(\overline{K_{l-2}} + K_{2l+1,2l+1} \subseteq G \subseteq K_{l-2} + K_{2l+1,2l+1}\), \(G\) contains \(l – 1\) vertex-disjoint \(5\)-cycles and a path of order \(5\), which is vertex-disjoint from the \(l – 1\) \(5\)-cycles. In fact, we prove a more general result that if \(G\) is a graph of order \(5k + 2s\) with \(\omega_2(G) \geq 6k + 2s\), then, unless \(\overline{K_{k}} + K_{2k+s,2k+s} \subseteq G \subseteq K_{k} + K_{2k+s,2k+s}\), \(G\) contains \(k+1\) vertex-disjoint \(5\)-cycles and a path of order \(2s – 5\), which is vertex-disjoint from the \(k + 1\) \(5\)-cycles. As an application of this theorem, we give a short proof for determining the exact value of \(\text{ex}(n,(k + 1)C_5)\), and characterize the extremal graph.

In this paper, we present the complex factorizations of the Jacobsthal and Jacobsthal Lucas numbers by determinants of tridiagonal matrices.

In this paper, we find families of \((0, -1, 1)\)-tridiagonal matrices whose determinants and permanents equal the negatively subscripted Fibonacci and Lucas numbers. Also, we give complex factorizations of these numbers by the first and second kinds of Chebyshev polynomials.

We classify all finite near hexagons which satisfy the following properties for a certain \(t_2 \in \{1,2,4\}\):(i) every line is incident with precisely three points;(ii) for every point \(x\), there exists a point \(y\) at distance \(3\) from \(x\);(iii) every two points at distance \(2\) from each other have either \(1\) or \(t_2 + 1\) common neighbours;(iv) every quad is big. As a corollary, we obtain a classification of all finite near hexagons satisfying (i), (ii) and (iii) with \(t_2\) equal to \(4\).

In this paper, we obtain the largest Laplacian spectral radius for bipartite graphs with given matching number and use them to characterize the extremal general graphs.

For integers \(k, \theta \leq 3\) and \(\beta \geq 1\), an integer \(k\)-set \(S\) with the smallest element \(0\) is a \((k; \beta, \theta)\)-free set if it does not contain distinct elements \(a_{i,j}\) (\(1 \leq i \leq j \leq \theta\)) such that \(\sum_{j=1}^{\theta -1}a_{i ,j} = \beta a_{i_\theta}\). The largest integer of \(S\) is denoted by \(\max(S)\). The generalized antiaverage number \(\lambda(k; \beta, \theta)\) is equal to \(\min\{\max(S) : S \text{ is a } (k^0; \delta, 0)\text{-free set}\}\). We obtain:(1) If \(\beta \notin \{\theta-2, \theta-1, \theta\}\), then \(\lambda(m; \beta, \theta) \leq (\theta-1)(m-2) + 1\); (2) If \(\beta \geq {\theta-1}\), then \(\lambda(k; \beta, \theta) \leq \min\limits_{k=m+n}\{\lambda(m;\beta,\theta)+\beta \lambda (n;\beta,\theta)+1\}\), where \(k =m+n \) with \(n>m\geq 3\) and \(\lambda(2n;\beta,\theta)\leq \lambda(n;\beta,\theta)(\beta+1)+\varepsilon\), for \(\varepsilon=1\) for \(\theta=3\) and \(\varepsilon=0\) otherwise.

A connected graph is highly irregular if the neighbors of each vertex have distinct degrees. We will show that every highly irregular tree has at most one nontrivial automorphism. The question that motivated this work concerns the proportion of highly irregular trees that are asymmetric, i.e., have no nontrivial automorphisms. A \(d\)-tree is a tree in which every vertex has degree at most \(d\). A technique for enumerating unlabeled highly irregular \(d\)-trees by automorphism group will be described for \(d \geq 4\) and results will be given for \(d = 4\). It will be shown that, for fixed \(d\), \(d \geq 4\), almost all highly irregular \(d\)-trees are asymmetric.

Combining with specific degrees or edges of a graph, this paper provides some new classes of upper embeddable graphs and extends the results in [Y. Huang, Y. Liu, Some classes of upper embeddable graphs, Acta Mathematica Scientia, \(1997, 17\)(Supp.): \(154-161\)].

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral trees \(S(r;m_i) = S(a_1+a_2+\cdots+a_s;m_1,m_2,\ldots,m_s)\) of diameter \(4\) with \(s = 2,3\). We give a better sufficient and necessary condition for the tree \(S(a_1+a_2;m_1,m_2)\) of diameter \(4\) to be integral, from which we construct infinitely many new classes of such integral trees by solving some certain Diophantine equations. These results are different from those in the existing literature. We also construct new integral trees \(S(a_1+a_2+a_3;m_1,m_2,m_3) = S(a_1+1+1;m_1,m_2,m_3)\) of diameter \(4\) with non-square numbers \(m_2\) and \(m_3\). These results generalize some well-known results of P.Z. Yuan, D.L. Zhang \(et\) \(al\).

Zagreb indices are the best known topological indices which reflect certain structural features of organic molecules. In this paper we point out that the modified Zagreb indices are worth studying and present some results about product graphs.

Let \(g \in H(\mathcal{B})\), \(g(0) = 0\) and \(\varphi\) be a holomorphic self-map of the unit ball \(\mathbb{B}\) in \(\mathbb{C}^n\). The following integral-type operator

\[I_\varphi^g(f)(z) = \int_{0}^{1} {\mathcal{R}f(\varphi(tz))}{g(tz)}\frac{ dt}{t}, \quad f \in H(\mathbb{B}),z\in \mathbb{B},\]

was recently introduced by S. Stević and studied on some spaces of holomorphic functions on \(\mathbb{B}\), where \(\mathcal{R}f(z) = \sum_{k=1}^n z_k \frac{\partial f}{\partial z_k}(z)\) is the radial derivative of \(g\). The boundedness and compactness of this operator from generally weighted Bloch spaces to Bloch-type spaces on \(\mathbb{B}\) are investigated in this note.

We start by proving that the Henson graphs \(H_n\), \(n \geq 3\) (the homogeneous countable graphs universal for the class of all finite graphs omitting the clique of size \(n\)), are retract rigid. On the other hand, we provide a full characterization of retracts of the complement of \(H_3\). Further, we prove that each countable partial order embeds in the natural order of retractions of the complements of Henson graphs. Finally, we show that graphs omitting sufficiently large null subgraphs omit certain configurations in their endomorphism monoids.

Combining integration method with series rearrangement,we establish several closed formulae for Gauss hypergeometric series with four free parameters, which extend essentially the related results found recently by Elsner \((2005).\)

In this paper, we study the global behavior of the nonnegative equilibrium points of the difference equation

\(x_{n+1} = \frac{Ax_{n-2l}}{B+C \prod\limits_{i=0}^{2k}x_{n-i}}, n=0,1,\ldots ,\)

where \(A\), \(B\), \(C\) are nonnegative parameters, initial conditions are nonnegative real numbers, and \(k\), \(l\) are nonnegative integers, \(l \leq k\). Also, we derive solutions of some special cases of this equation.

In this paper, the critical group structure of the Cartesian product graph \(C_4 \times C_n\) is determined, where \(n \geq 3\).

Let \(G = (V, E)\) be a simple connected graph with \(7\) vertices. The degree of \(v_i \in V\) and the average of degrees of the vertices adjacent to \(v_i\) are denoted by \(d_i\) and \(m_i\), respectively. The spectral radius of \(G\) is denoted by \(\rho(G)\). In this paper, we introduce a parameter into an equation of adjacency matrix, and obtain two inequalities for upper and lower bounds of spectral radius. By assigning different values to this parameter, one can obtain some new and existing results on spectral radius. Specially, if \(G\) is a nonregular graph, then

\[\rho(G) \leq \max_{1 \leq j<i \leq n} \{ \frac{d_i m_i – d_j m_j + \sqrt{(d_i m_i – d_j m_j)^2 – 4d_i d_j(d_i-d_j) (m_i – m_j)}}{2(d_i-d_j)} \}\] and \[\rho(G)\geq \min_{1 \leq j<i \leq n} \{ \frac{d_i m_i – d_j m_j + \sqrt{(d_i m_i – d_j m_j)^2 – 4d_i d_j(d_i-d_j) (m_i – m_j)}}{2(d_i-d_j)} \}.\] If \(G\) is a bidegreed graph whose vertices of same degree have equal average of degrees, then the equality holds.

An orientation of a simple graph \(G\) is called an oriented graph. If \(D\) is an oriented graph, \(\delta(D)\) its minimum degree and \(\lambda(D)\) its edge-connectivity, then \(\lambda(D) \leq \delta(D)\). The oriented graph is called maximally edge-connected if \(\lambda(D) = \delta(D)\) and super-edge-connected, if every minimum edge-cut is trivial. In this paper, we show that an oriented graph \(D\) of order \(n\) without any clique of order \(p + 1\) in its underlying graph is maximally edge-connected when

\[n \leq 4{\lfloor\frac{p\delta(D)}{p – 1}\rfloor}-1.\]

Some related conditions for oriented graphs to be super-edge-connected are also presented.

Denote by \(\mathcal{A_n}\), the set of the polyphenyl chains with \(n\) hexagons. For any \(A_n \in \mathcal{A_n}\), let \(m_k(A_n)\) and \(i_k(A_n)\) be the numbers of \(k\)-matchings and \(k\)-independent sets of \(A_n\), respectively. In the paper, we show that for any \(A_n \in \mathcal{A_n}\) and for any \(k \geq 0\),\(m_k(M_n) \leq m_k(A_n) \leq m_k(O_n) \quad \text{and} \quad i_k(M_n) \geq i_k(A_n) \geq i_k(O_n),\) with the equalities holding if \(A_n = M_n\) or \(A_n = O_n\), where \(M_n\) and \(O_n\) are the meta-chain and the ortho-chain, respectively. These generalize some related results in \([1]\).

Let \(G = (X, Y, E(G))\) be a bipartite graph with vertex set \(V(G) = X ! Y\) and edge set \(E(G)\), and let \(g, f\) be two nonnegative integer-valued functions defined on \(V(G)\) such that \(g(x) \leq f(x)\) for each \(x \in V(G)\). A \((g, f)\)-factor of \(G\) is a spanning subgraph \(F\) of \(G\) such that \(g(x) \leq d_F(x) \leq f(x)\) for each \(x \in V(F)\); a \((g, f)\)-factorization of \(G\) is a partition of \(E(G)\) into edge-disjoint \((g, f)\)-factors. Let \(\mathcal{F} = \{F_1, F_2, \ldots, F_m\}\) be a factorization of \(G\) and \(H\) be a subgraph of \(G\) with \(m\) edges. If \(F_i\), \(1 \leq i \leq m\), has exactly \(r\) edges in common with \(H\), we say that \(F_i\) is \(r\)-orthogonal to \(H\). In this paper, it is proved that every bipartite \((0, mf-(m-1)r)\)-graph has \((0, f)\)-factorizations randomly \(r\)-orthogonal to any given subgraph with \(m\) edges if \(2r \leq f(x)\) for any \(x \in V(G)\).

We define an \(r\)-capacitated dominating set of a graph \(G = (V,E)\) as a set \(\{v_1, \ldots, v_k\} \subseteq V\) such that there is a partition \((V_1, \ldots, V_k)\) of \(V\) where for all \(i\), \( v_i \in V_i\), \(v_i\) is adjacent to all of \(V_i – \{v_i\}\), and \(|V_i| \leq r + 1\). \(\daleth_r(G)\) is the minimum cardinality of an \(r\)-capacitated dominating set. We show properties of \(\daleth_r\), especially as regards the trivial lower bound \(|V|/(r + 1)\). We calculate the value of the parameter in several graph families, and show that it is related to codes and polyominoes. The parameter is \(NP\)-complete in general to compute, but a greedy approach provides a linear-time algorithm for trees.

On the basis of joint trees introduced by Yanpei Liu, by choosing different spanning trees and classifying the associated surfaces, we obtain the explicit expressions of genus polynomials for three types of graphs, namely \(K_5^n, W_6^n\) and \(K_{3,3}^n\), which are different from the graphs whose embedding distributions by genus have been obtained. And \(K_5^n\) and \(K_{3,3}^n\) are non-planar.

We develop the necessary machinery in order to prove that hexagonal tilings are uniquely determined by their Tutte polynomial, showing as an example how to apply this technique to the toroidal hexagonal tiling.

A \((d,1)\)-totel labelling of a graph \(G\) is an assignment of integers to \(V(G) \cap E(G)\) such that: (i) any two adjacent vertices of \(G\) receive distinct integers, (ii) any two adjacent edges of \(G\) receive distinct integers, and (iii) a vertex and its incident edge receive integers that differ by at least \(d\) in absolute value. The span of a \((d,1)\)-total labelling is the maximum difference between two labels. The minimum span of labels required for such a \((d, 1)\)-total labelling of \(G\) is called the \((d, 1)\)-total number and is denoted by \(\lambda_d^T(G)\). In this paper, we prove that \(\lambda_d^T(G)\geq d+r+1 \) for \(r\)-regular nonbipartite graphs with \(d \geq r \geq 3\) and determine the \((d, 1)\)-total numbers of flower snarks and of quasi flower snarks.

Let \(G = (V,E)\) be a simple graph with the vertex set \(V\) and the edge set \(E\). \(G\) is a sum graph if there exists a labelling \(f\) of the vertices of \(G\) into distinct positive integers such that \(uv \in E\) if and only if \( f(w)=f(u) + f(v) \) for some vertex \(w \in V\). Such a labelling \(f\) is called a sum labelling of \(G\). The sum number \(\sigma(G)\) of \(G\) is the smallest number of isolated vertices which result in a sum graph when added to \(G\). Similarly, the integral sum graph and the integral sum number \(\zeta(G)\) are also defined. The difference is that the labels may be any distinct integers.
In this paper, we will determine that
\[\begin{cases}
0 = \zeta(\overline{P_4}) < \sigma(\overline{P_4}) = 1;\\ 1 = \zeta(\overline{P_5}) < \sigma(\overline{P_5}) = 2;\\ 3 = \zeta(\overline{P_6}) < \sigma(\overline{P_6}) = 4;\\ \zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 0, \text{ for } n = 1, 2, 3;\\ \zeta(\overline{P_n}) = \sigma(\overline{P_n}) = 2n – 7, \text{ for } n \geq 7; \end{cases}\] and \[\begin{cases} 0 = \zeta(\overline{F_5}) < \sigma(\overline{F_5}) = 1;\\ 2 = \zeta(\overline{F_5}) < \sigma(\overline{F_6}) = 2;\\ \zeta(\overline{F_c}) = \sigma(\overline{F_n}) = 0, \text{ for } n =3,4;\\ \zeta(\overline{F_n}) = \sigma(\overline{F_n}) = 2n – 8, \text{ for } n \geq 7. \end{cases}\]

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. In this paper, we study the PI indices of bicyclic graphs whose cycles do not share two or more common vertices.