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.

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.

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.

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

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.

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.

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.

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.

The cycles approach

p-Periodic nets can be derived from a voltage graph G with voltages in Z(p), the free abelian group of rank p, if the cyclomatic number γ of G is larger than p. Equivalently, one may describe a net by providing a set of (γ - p) cycle vectors of G forming a basis of the subspace of the cycle space of G with zero net voltage. Let M be the matrix of this basis expressed in the edge basis of the 1-chain space of G. A net is called totally unimodular whenever every sub-determinant of M belongs to the set {-1, 0, 1}. Only a finite set of totally unimodular nets can be derived from some finite graph. It is shown that totally unimodular nets are stable under the operation of edge-lattice deletion in a sense that makes them comparable to minimal nets. An algorithm for the complete determination of totally unimodular nets derived from some finite graph is presented. As an application, the full list of totally unimodular nets derived from graphs of cyclomatic numbers 3 and 4, without bridges, is given. It is shown that many totally unimodular nets frequently occur in crystal structures.

We consider undirected simple connected graphs. A bridge in a connected graph is an edge whose deletion makes the graph disconnected. A smooth graph is a graph without endpoints. Obviously, the set of graphs without bridges is a subset of the set of smooth graphs. Smooth graphs were enumerated by Wright [1]. Hanlon and Robinson [2] enumerated both labeled and unlabeled graphs without bridges. They obtained a functional equation and a nonlinear partial differential equation for the generating function of labeled graphs. However, their formulas are awkward and have not been reduced to a form convenient for computations. Apparently, that is why Sloane’s famous “On-Line Encyclopedia of Integer Sequences” [3] only provides data on the number of unlabeled graphs without bridges. The aim of the present paper is to obtain closed-form expressions for the number of labeled biand tricyclic graphs without bridges as well as the corresponding asymptotics for the number of such graphs as the number of vertices tends to infinity. Recall that a cutpoint of a connected graph is a vertex whose deletion (the edges incident to the vertex are deleted as well) makes the graph disconnected. A block is a connected graph without cutpoints or a maximal nontrivial connected subgraph without cutpoints [4, p. 200]. A graph is said to be uni-, bi-, or tricyclic if its cyclomatic number is 1, 2, or 3, respectively. For k ≥ 0, we denote the number of labeled blocks with n vertices and n+ k edges, the number of labeled connected graphs without bridges with n vertices and n+ k edges, and the number of labeled smooth graphs with n vertices and n+ k edges by u(n, n+ k), l(n, n+ k), and v(n, n + k), respectively. The cyclomatic number of any of these graphs is k + 1. Let Uk(w) and Lk(w) be the exponential generating functions,

If by s k is denoted the number of independent sets of cardinality k in a graph G, then $${I(G;x)=s_{0}+s_{1}x+\cdots+s_{\alpha}x^{\alpha}}$$ is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97---106, 1983), where ? = ?(G) is the size of a maximum independent set. The inequality |I (G; ?1)| ≤ 2 ?(G), where ?(G) is the cyclomatic number of G, is due to (Engstrom in Eur. J. Comb. 30:429---438, 2009) and (Levit and Mandrescu in Discret. Math. 311:1204---1206, 2011). For ?(G) ≤ 1 it means that $${I(G;-1)\in\{-2,-1,0,1,2\}.}$$ In this paper we prove that if G is a unicyclic well-covered graph different from C 3, then $${I(G;-1)\in\{-1,0,1\},}$$ while if G is a connected well-covered graph of girth ? 6, non-isomorphic to C 7 or K 2 (e.g., every well-covered tree ? K 2), then I (G; ?1) = 0. Further, we demonstrate that the bounds {?2 ?(G), 2 ?(G)} are sharp for I (G; ?1), and investigate other values of I (G; ?1) belonging to the interval [?2 ?(G), 2 ?(G)].

Graphs with cyclomatic number two having panconnected square

The Sturm–Liouville equations on the edges of a metric connected graph together with the boundary and matching conditions at the vertices generate a spectral problem for a self-adjoint operator. It is shown that if the graph is not cyclically connected, then the maximal multiplicity of an eigenvalue of such an operator is μ + gT − pTin, where μ is the cyclomatic number of the graph, and gT and pTin are the number of edges and the number of interior vertices, respectively, for the tree obtained by contracting all the cycles of the graph into vertices. If the graph is cyclically connected, then the maximal multiplicity of an eigenvalue is μ + 1.

On the spectral radius of quasi- k-cyclic graphs

Let G:=(V,E) be a simple graph; for I⊆V we denote by l(I) the number of components of G[I], the subgraph of G induced by I. For V1,…,Vn subsets of V, we define a function β(V1,…,Vn) which is expressed in terms of l(⋃ni=1Vi) and l(Vi∪Vj) for i≤j. If V1,…,Vn are pairwise disjoint independent subsets of V, the number β(V1,…,Vn) can be computed in terms of the cyclomatic numbers of G[⋃ni=1Vi] and G[Vi∪Vj] for i≠j. In the general case, we prove that β(V1,…,Vn)≥0 and characterize when β(V1,…,Vn)=0. This special case yields a formula expressing the length of members of an interval algebra \cite{s} as well as extensions to pseudo-tree algebras. Other examples are given.

Let $G:=(V,E)$ be a simple graph; for $I\subseteq V$ we denote by $l(I)$ the number of components of $G[I]$, the subgraph of $G$ induced by $I$. For $V_1,\ldots , V_n$ subsets of $V$, we define a function $\beta (V_1,\ldots , V_n)$ which is expressed in terms of $l\left(\bigcup _{i=1} ^{n} V_i\right)$ and $l(V_i\cup V_j)$ for $i\leq j$. If $V_1,\ldots , V_n$ are pairwise disjoint independent subsets of $V$, the number $\beta (V_1,\ldots , V_n)$ can be computed in terms of the cyclomatic numbers of $G\left[\bigcup _{i=1} ^{n} V_i\right]$ and $G[ V_i\cup V_j]$ for $i\neq j$. In the general case, we prove that $\beta (V_1,\ldots , V_n)\geq 0$ and characterize when $\beta (V_1,\ldots , V_n)= 0$. This special case yields a formula expressing the length of members of an interval algebra \cite{s} as well as extensions to pseudo-tree algebras. Other examples are given.

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A Roman k-dominating function on G is a function f : V (G) <TEX>$\rightarrow$</TEX> {0, 1, 2} such that every vertex u for which f(u) = 0 is adjacent to at least k vertices <TEX>$\upsilon_1,\;\upsilon_2,\;{\ldots},\;\upsilon_k$</TEX> with <TEX>$f(\upsilon_i)$</TEX> = 2 for i = 1, 2, <TEX>$\ldot$</TEX>, k. The weight of a Roman k-dominating function is the value f(V (G)) = <TEX>$\sum_{u{\in}v(G)}$</TEX> f(u). The minimum weight of a Roman k-dominating function on a graph G is called the Roman k-domination number <TEX>${\gamma}_{kR}$</TEX>(G) of G. Note that the Roman 1-domination number <TEX>$\gamma_{1R}$</TEX>(G) is the usual Roman domination number <TEX>$\gamma_R$</TEX>(G). In this paper, we investigate the properties of the Roman k-domination number. Some of our results extend these one given by Cockayne, Dreyer Jr., S. M. Hedetniemi, and S. T. Hedetniemi [2] in 2004 for the Roman domination number.

Tricyclic graphs with maximum Merrifield–Simmons index

Packing edge-disjoint cycles in graphs and the cyclomatic number