Cyclomatic Number Research Articles

Graphs with fixed number of pendent vertices and minimal Zeroth-order general Randić index

Graphs with fixed number of pendent vertices and minimal Zeroth-order general Randić index

Extremal Degree-Product Indices of Graphs with Fixed Number of Pendant Vertices and Cyclomatic Number

The Narumi-Katayama index and first multiplicative Zagreb index of a graph are defined as the product of the degrees of the vertices of and the product of square of the degrees of the vertices of , respectively. The second multiplicative Zagreb index is defined as . In this paper, we compute the extremal , and for the graphs with given order, number of pendant vertices and cyclomatic number.

The unicyclic graphs with maximum degree resistance distance

The unicyclic graphs with maximum degree resistance distance

Rainbow Numbers for Graphs Containing Small Cycles

For a given graph $$H$$H and $$n \ge 1,$$n?1, let $$f(n, H)$$f(n,H) denote the maximum number $$m$$m for which it is possible to colour the edges of the complete graph $$K_n$$Kn with $$m$$m colours in such a way that each subgraph $$H$$H in $$K_n$$Kn has at least two edges of the same colour. Equivalently, any edge-colouring of $$K_n$$Kn with at least $$rb(n,H)=f(n,H) + 1$$rb(n,H)=f(n,H)+1 colours contains a rainbow copy of $$H.$$H. The numbers $$f(n,H)$$f(n,H) and $$rb(K_n,H)$$rb(Kn,H) are called anti-ramsey numbers and rainbow numbers, respectively. In this paper we will classify the rainbow number for a given graph $$H$$H with respect to its cyclomatic number. Let $$H$$H be a graph of order $$p \ge 4$$p?4 and cyclomatic number $$v(H) \ge 2.$$v(H)?2. Then $$rb(K_n, H)$$rb(Kn,H) cannot be bounded from above by a function which is linear in $$n.$$n. If $$H$$H has cyclomatic number $$v(H) = 1,$$v(H)=1, then $$rb(K_n, H)$$rb(Kn,H) is linear in $$n.$$n. Moreover, we will compute all rainbow numbers for the bull $$B,$$B, which is the unique graph with $$5$$5 vertices and degree sequence $$(1,1,2,3,3)$$(1,1,2,3,3).

Recovering a quantum graph spectrum from vertex data

We study the question to what extent spectral information on a Schrödinger operator on a finite, compact metric graph subject to standard or δ-type matching conditions can be recovered from a corresponding Titchmarsh–Weyl function on the boundary of the graph. In contrast to the case of ordinary or partial differential operators, the knowledge of the Titchmarsh–Weyl function is in general not sufficient for recovering the complete spectrum of the operator (or the potentials on the edges). However, it is shown that those eigenvalues with sufficiently high (depending on the cyclomatic number of the graph) multiplicities can be recovered. Moreover, we prove that under certain additional conditions the Titchmarsh–Weyl function even contains the full spectral information.

Graphs with fixed number of pendent vertices and minimal first Zagreb index

The rst Zagreb index M1 of a graph G is equal to the sum of squares of degrees of the vertices of G. Goubko proved that for trees with n1 pendent vertices, M1 9n1 16. We show how this result can be extended to hold for any connected graph with cyclomatic number 0. In addition, graphs with n vertices, n1 pendent vertices, cyclomatic number , and minimal M1 are characterized. Explicit expressions for minimal M1 are given for = 0;1;2, which directly can be extended for > 2.

On a conjecture of Randic index and graph radius

The Randi\'{c} index $R(G)$ of a graph $G$ is defined as the sum of $(d_i d_j)^{-1/2}$ over all edges $v_i v_j$ of $G$, where $d_i$ is the degree of the vertex $v_i$ in $G$. The radius $r(G)$ of a graph $G$ is the minimum graph eccentricity of any graph vertex in $G$. Fajtlowicz(1988) conjectures $R(G) \ge r(G)-1$ for all connected graph $G$. A stronger version, $R(G) \ge r(G)$, is conjectured by Caporossi and Hansen(2000) for all connected graphs except even paths. In this paper, we make use of Harmonic index $H(G)$, which is defined as the sum of $\frac{2}{d_i+d_j}$ over all edges $v_i v_j$ of $G$, to show that $R(G) \ge r(G)-31/105(k-1)$ for any graph with cyclomatic number $k\ge 1$, and $R(T)> r(T)+1/15$ for any tree except even paths. These results improve and strengthen the known results on these conjectures.

Characterization of Jos City Road Network, Nigeria

ἀe performance of road network depends on its topological characteristics which help to deḀne its connectiv-ity. ἀis paper analyses the topological characteristic of Jos city road network and its bearing on traᴀc ᰀow situations. Simple graph theoretic measures oᬀered the framework on which the problem was approached. ἀe study requires the abstraction and analysis of the topological structure by selection of certain variables relating to the road connectivity. ἀese include the Beta, Gamma and Alpha index, the PI, Cyclomatic number, and the spread and density of the network. Information on these variables was obtained through the use of vector data model to abstract the road network graph from the Quick-bird satellite imagery used for the study. Results of the Ḁndings reveal that, the road network of Jos City Centre as a whole have achieved an average level of connectivity, showing Beta index values of 1.4049, Gamma index value of 47.06%, Alpha index Value of 20.63%; and a pi and cyclomatic number of 24.74 and 165 respectively, the spread of the network is moderate exhibiting a value of 23, even though some areas have more concentration of roads than the others; and has a road density of 52 links per km2. Based on these Ḁndings, the need for construction of new roads is imperative so as to improve the eᴀciency of connectivity and accessibility within the city.

On graphs with many cycles

On graphs with many cycles

On a new cyclicity measure of graphs—The global cyclicity index

On a new cyclicity measure of graphs—The global cyclicity index

A spatio-temporal modelling approach for the study of the connectivity and accessibility of the Guangzhou metropolitan network

A spatio-temporal modelling approach for the study of the connectivity and accessibility of the Guangzhou metropolitan network

On a new cyclicity measure of graphs—The global cyclicity index

On a new cyclicity measure of graphs—The global cyclicity index

The Partial Inverse Minimum Connected Spanning Subgraph Problem with Given Cyclomatic Number

In this paper, we consider a partial inverse problem of minimum connected spanning subgraph with cyclomatic number . That is, given a subgraph, a cyclomatic number and a constraint that the edge weights can only decrease, we want to modify the edge weights as little as possible, so that there exists a minimum connected spanning subgraph with cyclomatic number and containing the given subgraph. For the case that the given subgraph is a connected subgraph with cyclomatic number , we solve the problem in polynomial time by contracting the subgraph. And when the given subgraph is a spanning tree, we solve the problem in polynomial time by using the MST algorithm.

Wiener Index of Graphs and Their Line Graphs

The Wiener index W(G) of a graph G is a distance-based topological index defined as the sum of distances between all pairs of vertices in G. It is shown that for λ=2 there is an infinite family of planar bipartite chemical graphs G of girth 4 with the cyclomatic number λ, but their line graphs are not chemical graphs, and for λ⩾2 there are two infinite families of planar nonbipartite graphs G of girth 3 with the cyclomatic number λ; the three classes of graphs have the property W(G)=W(L(G)), where L(G) is the line graph of G.

О некоторых свойствах n-последовательносвязной цепи

It is introduced the class of the nonorientedn-sequentially connected chain. It is considered the applicationfields of then-sequentially connected chains, in particular the problems of optimal location in discrete formulations, and the problems of selection the optimal conduct in systems, described by Markov controllable processes. The main characteristics ofn-sequentially connected chains, such as the number of edges, the size of the maximum clique, the chromatic and the cyclomatic numbers, etc. are given. The relations of the classn-sequentially connected chains to perfect, triangulated, composite and splittable classes of graphs are determined.

Remarks on a conjecture about Randić index and graph radius

LetG be a nontrivial connected graph. The radiusr.G/ ofG is the minimum eccent- ricity among eccentricities of all vertices in G. The Randiindex of G is defined as R.G/D P uv2E.G/ 1 p d G.u/dG.v/ , and the Harmonic index is defined asH.G/D P uv2E.G/ 2 dG.u/CdG.v/ , where dG.x/ is the degree of the vertex x in G. In 1988, Fajtlowicz conjectured that for any connected graphG,R.G/ r.G/ 1. This conjecture remains still open so far. More recently, Deng et al. proved that this conjecture is true for connected graphs with cyclomatic number no more than 4 by means of Harmonic index. In this short paper, we use a class of composite gra- phs to construct infinite classes of connected graphs, with cyclomatic number greater than 4, for which the above conjecture holds. In particular, for any given positive odd number k 7, we construct a connected graph with cyclomatic numberk such that the above conjecture holds for this graph. 2010 Mathematics Subject Classification: 05C07; 05C12; 05C76

Fractal Properties of Linux Kernel Maps

Many different measures were proposed to describe the problem of possible software complexity - the number of lines of code sometimes referred as a source lines of code (SLOC), Halstead’s volume V, McCabe cyclomatic number V(G), among others. However, any of them doesn’t take into account the possible fractal properties of software source code emerging from development process. The main aim of this paper is to show that in the case of successive Linux OS kernels fractal self-organization of the system can be seen. This is done in the relation to: (i) the analysis of rate of growth for number of files and source lines of Linux kernels code, (ii) by the presentation of some visualizations indicating self-similar graphical structure of OS kernels, (iii) by the calculations of fractal dimensions Db basing on box dimension method. Basing on obtained results it can be assumed that: (i) calculated rate of growth in the case of lines and files in the simplest approach can be approximated by the polynomial with degree 2 with R = 0.96 and R = 0.94 respectively, (ii) this system becomes more and more complex with self-similar structure, (iii) its fractal dimension is still growing. Presented analysis opens new possibilities for description of computer programs in terms of complex systems approach.

Graphs with many vertex-disjoint cycles

Graph Theory We study graphs G in which the maximum number of vertex-disjoint cycles nu(G) is close to the cyclomatic number mu(G), which is a natural upper bound for nu(G). Our main result is the existence of a finite set P(k) of graphs for all k is an element of N-0 such that every 2-connected graph G with mu(G)-nu(G) = k arises by applying a simple extension rule to a graph in P(k). As an algorithmic consequence we describe algorithms calculating minmu(G)-nu(G), k + 1 in linear time for fixed k.

웹 프로그래밍을 위한 복잡도 한계값의 적정성

본 연구는 웹 환경에서 어플리케이션 복잡도의 빈도분포를 근거로 한계값의 적정성을 분석하기 위한 실험을 하였으며 두 가지 가정을 기준으로 작업하였다. 즉, 절차적 프로그래밍에서 McCabe의 상한값 10과 자바 프로그래밍에서의 Lopez의 상한값 5에 대하여 웹 프로그래밍 구문에 이들 설정값의 적용이 가능한가?에 대한 실험으로 10 웹 사이트 프로젝트를 수집하였고 4,000여개의 ASP파일 표본이 측정되었다. 웹 어플리케이션에 대한 복잡도 빈도분포를 파악한 결과 통합된 웹 어플리케이션의 90% 이상이 복잡도 50이하의 값을 가짐으로써 한계값 50이 제안되었다. 서버, 클라이언트, HTML이 통합된 웹 어플리케이션의 구조상 HTML의 복잡도가 35~40의 값을 가지게 되는데 이는 HTML이 주로 홈 페이지나 사이트 맵을 구성하는 메뉴 형태로 되어 있어 높은 복잡도의 적합성이 설명되었다. 향후 웹 어플리케이션의 구조상 복잡도와 관련된 숨어 있는 속성은 없는지 관련성을 찾아보는 노력이 필요하다. In this empirical study at the Web environment based on the frequency distribution of the cyclomatic complexity number of the application, the relevance of the threshold has been analyzed with the next two assumptions. The upper bound established by McCabe in the procedural programming equals 10 and the upper bound established by Lopez in the Java programming equals 5. Which numerical value can be adapted to Web application contexts? In order to answer this 10 web site projects have been collected and a sample of more than 4,000 ASP files has been measured. After analyzing the frequency distribution of the cyclomatic complexity of the Web application, experiment result is that more than 90% of Web application have a complexity less than 50 and also 50 is proposed as threshold of Web application. Web application has the complex architecture with Server, Client and HTML, and the HTML side has the high complexity 35~40. The reason of high complexity is that HTML program is usually made of menu type for home page or site map, and the relevance of that has been explained. In the near future we need to find out if there exist some hidden properties of the Web application architecture related to complexity.

Combinatorial Algorithm for Finding Spanning Forests

In this paper we present a new combinatorial algorithm for finding all different spanning forests for a disconnected graph G, depending on the adjacency matrix, the cyclomatic number, the combination between the numbers of edges and the cyclomatic number, and the permutation between the entries of the adjacency matrix to determine the spanning forests.