Iterated Line Graphs Research Articles

Hamiltonian Indices of Three Classes of Graphs Obtained from Petersen Graph

In this paper, we mainly consider the Hamiltonian indices of three classes of graphs obtained from Petersen graph, that is, the minimum integer m of m-time iterated line graph Lm(G) of these three classes of graphs such that Lm(G) is Hamiltonian. We show that the Hamiltonian indices of those graphs obtained by replacing every vertex of Petersen graph with a n-cycle or a complete graph of order n, or adding n pendant edges to each vertex of Petersen graph are both 2. In addition, we also study the situations of adding an edge to these three classes of graphs and obtain that their Hamiltonian indices are both 2.

On the dominating (induced) cycles of iterated line graphs

In this paper, we give characterizations of graphs with line graphs or iterated line graphs that have dominating cycles. The characterization of graphs with dominating cycles in its line graphs and its i -iterated line graphs for i ≥ 2 are different: we may not unify them. As an application, we give characterizations of graphs with iterated line graphs that have dominating induced cycles. They are very different from the known results, although those characterizations for dominating cycles have some similarities with results on hamiltonian iterated line graphs of Harary and Nash-Williams (1965) and Xiong and Liu (2002). Using these results, we also give some analysis on the complexity of determining the existence of dominating cycles. It is NP-complete to decide whether a given graph has a dominating induced cycle, even for a 2-iterated line graph.

Bounds on the connectivity of iterated line graphs

For simple connected graphs that are neither paths nor cycles, we define l ( G )=max{ m : G has a divalent path of length m that is not both of length 2 and in a K 3 } , where a divalent path is a path whose internal vertices have degree two in G . Let G be a graph and L n ( G ) be its n -th iterated line graph of G . We use κ e ′( G ) and κ ( G ) for the essential edge connectivity and vertex connectivity of G , respectively. Let G be a simple connected graph that is not a path, a cycle or K 1, 3 , with l ( G )= l ≥ 1 . We prove that (i) for integers s ≥ 1 , κ ′ e ( L l + s ( G )) ≥ 2 s + 2 ; (ii) for integers s ≥ 2 , κ ( L l + s ( G )) ≥ 2 s − 1 + 2 . The bounds are best possible.

Asymptotically sharpening the $s$-Hamiltonian index bound

For a non-negative integer $s\le |V(G)|-3$, a graph $G$ is $s$-Hamiltonian if the removal of any $k\le s$ vertices results in a Hamiltonian graph. Given a connected simple graph $G$ that is not isomorphic to a path, a cycle, or a $K_{1,3}$, let $\delta(G)$ denote the minimum degree of $G$, let $h_s(G)$ denote the smallest integer $i$ such that the iterated line graph $L^{i}(G)$ is $s$-Hamiltonian, and let $\ell(G)$ denote the length of the longest non-closed path $P$ in which all internal vertices have degree 2 such that $P$ is not both of length 2 and in a $K_3$. For a simple graph $G$, we establish better upper bounds for $h_s(G)$ as follows. \begin{equation*} h_s(G)\le \left\{ \begin{aligned} & \ell(G)+1, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s=0;\\ & \widetilde d(G)+2+\lceil \lg (s+1)\rceil, &&\mbox{ if }\delta(G)\le 2 \mbox{ and }s\ge 1;\\ & 2+\left\lceil\lg\frac{s+1}{\delta(G)-2}\right\rceil, && \mbox{ if } 3\le\delta(G)\le s+2;\\ & 2, &&{\rm otherwise}, \end{aligned} \right. \end{equation*} where $\widetilde d(G)$ is the smallest integer $i$ such that $\delta(L^i(G))\ge 3$. Consequently, when $s \ge 6$, this new upper bound for the $s$-hamiltonian index implies that $h_s(G) = o(\ell(G)+s+1)$ as $s \to \infty$. This sharpens the result, $h_s(G)\le\ell(G)+s+1$, obtained by Zhang et al. in [Discrete Math., 308 (2008) 4779-4785].

Certain classes of complementary equienergetic graphs

The energy of a graph is the sum of the absolute values of the eigenvalues of a graph‎. ‎Two graphs are said to be equienergetic if they have same energy‎. ‎A graph is said to be complementary equienergetic if it is equienergetic with its complement‎. ‎Recently several complementary equienergetic graphs have been identified‎. ‎In this paper‎, ‎we characterize the cycles‎, ‎paths‎, ‎complete bipartite regular graphs and iterated line graphs of regular graphs‎, ‎which are complementary equienergetic‎. .

On Computing the Hamiltonian Index of Graphs ⋆

The $r$-th iterated line graph $L^{r}(G)$ of a graph $G$ is defined by: (i) $L^{0}(G) = G$ and (ii) $L^{r}(G) = L(L^{(r- 1)}(G))$ for $r > 0$, where $L(G)$ denotes the line graph of $G$. The Hamiltonian Index $h(G)$ of $G$ is the smallest $r$ such that $L^{r}(G)$ has a Hamiltonian cycle. Checking if $h(G) = k$ is NP-hard for any fixed integer $k \geq 0$ even for subcubic graphs $G$. We study the parameterized complexity of this problem with the parameter treewidth, $tw(G)$, and show that we can find $h(G)$ in time $O*((1 + 2^{(\omega + 3)})^{tw(G)})$ where $\omega$ is the matrix multiplication exponent and the $O*$ notation hides polynomial factors in input size. The NP-hard Eulerian Steiner Subgraph problem takes as input a graph $G$ and a specified subset $K$ of terminal vertices of $G$ and asks if $G$ has an Eulerian (that is: connected, and with all vertices of even degree.) subgraph $H$ containing all the terminals. A second result (and a key ingredient of our algorithm for finding $h(G)$) in this work is an algorithm which solves Eulerian Steiner Subgraph in $O*((1 + 2^{(\omega + 3)})^{tw(G)})$ time.

The planar and outerplanar indices of Cayley graphs of finite groups

In our study, we consider the Cayley graph of finite groups and its iterated line graphs. We present a complete characterization of finite groups with planar and outerplanar indices.

Degree sum conditions for hamiltonian index

In this note, we show a sharp lower bound of min left{{sumnolimits_{i = 1}^k {{d_G}({u_i}):{u_1}{u_2} ldots {u_k}}} right. is a path of (2-)connected G on its order such that (k-1)-iterated line graphs Lk−1(G) are hamiltonian.

Topological characteristics of iterated line graphs in the QSAR problem: a multigraph in the description of properties of unsaturated hydrocarbons

A recently tested (Zakharov et al., Journal of Chemometrics 33:e3169, 2019) iterated line graph approach to the description of the physico-chemical characteristics of hydrocarbons has been generalized to describe the properties of molecular systems with multiple bonds. The main idea of the new approach is to start with a regular molecular graph and generate a sequence of graphs where every next one is the line (edge) graph of the previous one. For systems with multiple bonds (e.g. alkenes, alkynes), the applicability of the iterated line graph approach has been demonstrated. Our approach made it possible to show that molecules with double and triple bonds have a corresponding representation with multigraphs and, consequently, a set of line graphs. It was demonstrated that a single topological index calculated for a graph and an iterated line graph sequence can be used for building a systematic set of regression equations with satisfactory predictability.

Nonplanarity of Iterated Line Graphs

The 1-crossing index of a graph G is the smallest integer k such that the k th iterated line graph of G has crossing number greater than 1. In this paper, we show that the 1-crossing index of a graph is either infinite or it is at most 5. Moreover, we give a full characterization of all graphs with respect to their 1-crossing index.

COHERENCE ANALYSIS FOR ITERATED LINE GRAPHS OF MULTI-SUBDIVISION GRAPH

More and more attention has focused on consensus problem in the study of complex networks. Many researchers investigated consensus dynamics in a linear dynamical system with additive stochastic disturbances. In this paper, we construct iterated line graphs of multi-subdivision graph by applying multi-subdivided-line graph operation. It has been proven that the network coherence can be characterized by the Laplacian spectrum of network. We study the recursion formula of Laplacian eigenvalues of the graphs. After that, we obtain the scalings of the first- and second-order network coherence.

A study of the generalized outerplanar index of zero-divisor graphs

The purpose of this paper is to explore the question of embedding a zero-divisor graph and its iterated line graphs on the plane such that the resulting graphs are generalized outerplanar graphs. We give a full characterization of all zero-divisor graphs with respect to their generalized outerplanar index.

Topological characteristics of iterated line graphs in QSAR problem: Octane numbers of saturated hydrocarbons

AbstractCapabilities of line (edge) graph in quantitative structure‐activity relationship (QSAR) problem for constructing predictive models are demonstrated. The basic idea for proposed approach is to describe molecular structure using generated graph sequence: vertex and chain of line graphs. Every following graph in the sequence corresponds to the structure where vertices appear to be edges of the previous one. For every graph, it is proposed to compute corresponding topological and informational descriptors. Employing latter for derivation of predictive equations allows to realize progressive hierarchy of models. As a test problem, the forecast of research octane numbers for saturated hydrocarbons is chosen. Among the descriptors selected to be used for building regression models well‐known Randic connectivity index (χ(1)), first Zagreb index (ZM1) and theoretic‐informational index of vertex complexity (Vc) are employed. Derived hierarchy of regression equations shows systematic improvement in the description of desired property as every new graph is included in consideration. Determination coefficients for training sample and computed via leave‐one‐out cross‐validation procedure reveal that four‐parametric equation with Vc descriptor has satisfactory forecasting facilities. Efficiency of proposed approach was also demonstrated on the example of artificial neural networks.

A new approach in topological descriptors usage. Iterated line graphs in the theoretical prediction of physico-chemical properties of saturated hydrocarbons

A new look on the problem of the molecular systems index description is presented. The capabilities of iterated line (edge) graphs in characterization of saturated hydrocarbons properties were investigated. It was demonstrated that single selected molecular (graph-theoretical (topological) or informational) descriptor calculated for the sequence of nested line graphs provides quite reliable progressive set of regression equations. Hence, the problem of descriptor set reduction is solved in the presented approach at list partially. Corresponding program complex (QUASAR) has been implemented with Python 3 program language. As the test example physico-chemical properties of octane isomers have been chosen. Among the properties under investigation there are boiling point, critical temperature, critical pressure, enthalpy of vaporization, enthalpy of formation, surface tension and viscosity. The corresponding rather simple linear regression equations which include one, two or three parameters correspondingly have been obtained. The predictive ability of the equations has been investigated using internal validation tests. The test by leave-one-out (LOO) validation and Y‑scrambling evaluate the obtained equations as adequate. For instance, for the regression model for boiling point the best equation characterizes by determination coefficients R2 = 0.943, with LOO procedure – Q2 = 0.918, while for the Y-scrambling test Q2y-scr<0.3 basically. It is shown that all the abovementioned molecular properties in iterated line graph approach can be effectively described by commonly used topological indices. Namely almost every randomly selected topological index can give adequate equation. Effectiveness is demonstrated on the example of Zagreb group indices. Also essential effectiveness and rather universal applicability of the so-called “forgotten” index (ZM3) was demonstrated.

Essential edge connectivity of line graphs

Let G be a graph and L(G) be its line graph. In 1969, Chartrand and Stewart proved that κ′(L(G))≥2κ′(G)−2, where κ′(G) and κ′(L(G)) denote the edge connectivity of G and L(G) respectively. We show a similar relationship holds for the essential edge connectivity of G and L(G), written κe′(G) and κe′(L(G)), respectively. In this note, it is proved that if L(G) is not a complete graph and G does not have a vertex of degree two, then κe′(L(G))≥2κe′(G)−2. An immediate corollary is that κ(L2(G))≥2κ(L(G))−2 for such graphs G, where the vertex connectivity of the line graph L(G) and the second iterated line graph L2(G) are written as κ(L(G)) and κ(L2(G)) respectively.

On a conjecture about the ratio of Wiener index in iterated line graphs

Let G be a graph. Denote by W(G) its Wiener index and denote by Li(G) its i-iterated line graph. Dobrynin and Mel’nikov proposed to estimate the extremal values for the ratio Rk(G) = W(Lk(G)) / W(G) for k ≥ 1. Motivated by this we study the ratio for higher k’s. We prove that among all trees on n vertices the path Pn has the smallest value of this ratio for k ≥ 3. We conjecture that this holds also for k = 2, and even more, for the class of all connected graphs on n vertices. Moreover, we conjecture that the maximum value of the ratio is obtained for the complete graph.

D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated line graphs of G are of diameter 2 and that ED(overline{Lk(G)}), k≥2 depends only on p and r. This result leads to the construction of regular D-equi energetic pair of graphs.

Panconnected index of graphs

For a connected graph G not isomorphic to a path, a cycle or a K1,3, let pc(G) denote the smallest integer n such that the nth iterated line graph Ln(G) is panconnected. A path P is a divalent path of G if the internal vertices of P are of degree 2 in G. If every edge of P is a cut edge of G, then P is a bridge divalent path of G; if the two ends of P are of degree s and t, respectively, then P is called a divalent(s,t)-path. Let ℓ(G)=max{m:G has a divalent path of length m that is not both of length 2 and in a K3}. We prove the following.(i) If G is a connected triangular graph, then L(G) is panconnected if and only if G is essentially 3-edge-connected.(ii) pc(G)≤ℓ(G)+2. Furthermore, if ℓ(G)≥2, then pc(G)=ℓ(G)+2 if and only if for some integer t≥3, G has a bridge divalent (3,t)-path of length ℓ(G).

Planarity and outerplanarity indexes of the zero-divisor graphs

In this paper, we consider the problem of planarity and outerplanarity of iterated line graphs of the zero divisor graphs for finite commutative rings. We give a full characterization of all zero divisor graphs with respect to their planarity and outerplanarity indexes.

Even factors with a bounded number of components in iterated line graphs

We consider even factors with a bounded number of components in the $n$-times iterated line graphs $L^{n}(G)$. We present a characterization of a simple graph $G$ such that $L^n(G)$ has an even factor with at most $k$ components, based on the existence of a certain type of subgraphs in $G$. Moreover, we use this result to give some upper bounds for the minimum number of components of even factors in $L^n(G)$ and also show that the minimum number of components of even factors in $L^n(G)$ is stable under the closure operation on a claw-free graph $G$, which extends some known results. Our results show that it seems to be NP-hard to determine the minimum number of components of even factors of iterated line graphs. We also propose some problems for further research.