Cyclomatic Number Research Articles

Inertia indices of a complex unit gain graph in terms of matching number

A complex unit gain graph is a triple (or for short) consisting of a simple graph G, as the underlying graph of , the set of unit complex numbers and a gain function such that . Let be the adjacency matrix of . In this paper, we prove that where , , and are the number of positive eigenvalues of , the number of negative eigenvalues of , the matching number and the cyclomatic number of G, respectively. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds, respectively.

Graphs with eigenvalue −1 of multiplicity 2θ(G)+ρ(G)−1

Let G be a graph, m(G,λ) be the multiplicity of eigenvalue λ of A(G), θ(G) be the cyclomatic number of G, ρ(G) be the number of pendent vertices of G. Wang et al. proved that m(G,λ)≤2θ(G)+ρ(G)−1 for an arbitrary eigenvalue λ if G is not a cycle, and left an open problem to characterize the extremal graphs attaining the upper bound. We solve the open problem when λ=−1, that is a graph G satisfies m(G,−1)=2θ(G)+ρ(G)−1 if and only if it is obtained from a tree T with m(T,−1)=2θ(T)+ρ(T)−1 by attaching θ(G) cycles of order a multiple of 3 to θ(G) quasi-pendent vertices of T and delete related θ(G) pendent vertices, where ρ(T)≥θ(G).

Evaluation of Connectivity Algorithms in Road Network Analysis of Oriire Local Government Area of Oyo State, Nigeria

Rural road network has been considered as the combinations of several routes into a more or less integrated structure permitting movements between many nodes. The study evaluated road network level of seven (7) villages in Oriire LGA of Oyo State. Oriire LGA is one of the agricultural producing areas of Oyo State this is because the major occupation is agriculture. Geographical information system was used to capture the road network pattern of the study area. Connectivity indices such as alpha (α), beta (β), gamma (γ) and cyclomatic number (μ) were adopted to evaluate the level of connectivity. The result revealed that, (α = 0.87), (β = 1.87), (γ = 0.78) and (μ = 7) which are significant as 0.5, 0.1, 0.5 and 1-10 respectively. These values indicated that, the selected villages have nodal connectivity of at least three (3) and that, the connectivity level is at medium. The study concluded that, the study area is connected by roads however, the connectivity do not tend to the most important village (Tewure) the rural commercial center. It is therefore suggested that, Onigba with the highest nodal connectivity be made a mini-rural market.

An analysis of the parameterized complexity of periodic timetabling

Public transportation networks are typically operated with a periodic timetable. The periodic event scheduling problem (PESP) is the standard mathematical modeling tool for periodic timetabling. PESP is a computationally very challenging problem: For example, solving the instances of the benchmarking library PESPlib to optimality seems out of reach. Since PESP can be solved in linear time on trees, and the treewidth is a rather small graph parameter in the networks of the PESPlib, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth, or even better, fixed-parameter tractable algorithms. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. Moreover, we show W[1]-hardness of the general feasibility problem with respect to treewidth, which means that we can most likely only accomplish pseudo-polynomial-time algorithms on input networks with bounded tree- or branchwidth. We present two such algorithms based on dynamic programming. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special—but standard—structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.

Graphs [formula omitted] with nullity [formula omitted

The nullity η(G) of a (simple) graph G is the multiplicity of 0 as an eigenvalue of the adjacency matrix of G. It was shown by Ma et al. (2016) that for a connected graph G with n(G) vertices, e(G) edges and p(G)(≥1) pendant vertices, η(G)≤2c(G)+p(G)−1, where c(G)=e(G)−n(G)+θ(G) is the cyclomatic number of G, θ(G) being the number of connected components of G. In this paper connected graphs G and trees T that satisfy respectively the conditions η(G)=2c(G)+p(G)−1 and η(T)=p(T)−1 are characterized. Besides, we identify the matched and mismatched vertices in a tree T that satisfies η(T)=p(T)−1 and characterize trees T that have a pendant vertex matched in T and satisfy η(T)=p(T)−2. We obtain new sharp lower and upper bounds for a tree T with p(T)≥3, given in terms of p(T) and the numbers of two special kinds of internal paths in T. It is also proved that for any integer c≥2, the set of values attained by η(G)−n(G)+2m(G), as G runs through all connected, leaf-free graphs G with cyclomatic number c, is {−c+1,−c+2,…,2c−3,2c−2}.

ON N-VERTEX CHEMICAL GRAPHS WITH A FIXED CYCLOMATIC NUMBER AND MINIMUM GENERAL RANDI´C INDEX

"The general Randi´c index of a graph G is defined as Rα(G) = P uv∈E(G)(dudv)α, where du and dv denote the degrees of the vertices u and v, respectively, α is a real number, and E(G) is the edge set of G. The minimum number of edges of a graph G whose removal makes G as acyclic is known as the cyclomatic number and it is usually denoted by ν. A graph with the maximum degree at most 4 is known as a chemical graph. For ν = 0, 1, 2 and α > 1, the problem of finding graph(s) with the minimum general Randi´c index Rα among all n-vertex chemical graphs with the cyclomatic number ν has already been solved. In this paper, this problem is solved for the case when ν ≥ 3, n ≥ 5(ν − 1), and 1 < α < α0, where α0 ≈ 11.4496 is the unique positive root of the equation 4(8α − 6α) + 4α − 9α = 0."

On the extremal graphs for Second Zagreb Index with fixed number of vertices and cyclomatic number

On the extremal graphs for Second Zagreb Index with fixed number of vertices and cyclomatic number

Bounds and extremal graphs of second reformulated Zagreb index for graphs with cyclomatic number at most three

Miliˇcevi´c et al., in 2004, introduced topological indices known as Reformulated Zagreb indices, where they modified Zagreb indices using the edge-degree instead of vertex degree. In this paper, we present a simple approach to find the upper and lower bounds of the second reformulated Zagreb index, EM2(G), by using six graph operations/transformations. We prove that these operations significantly alter the value of reformulated Zagreb index. We apply these transformations and identify those graphs with cyclomatic number at most 3, namely trees, unicyclic, bicyclic and tricyclic graphs, which attain the upper and lower bounds of second reformulated Zagreb index for graphs.

Eigenvalues, traceability, and perfect matching of a graph

It is well known that the recognition problem of the existence of a perfect matching in a graph, as well as the recognition problem of its Hamiltonicity and traceability, is NP-complete. Quite recently, lower bounds for the size and the spectral radius of a graph that guarantee the existence of a perfect matching in it have been obtained. We improve these bounds, firstly, by using the available bounds for the size of the graph for existence of a Hamiltonian path in it, and secondly, by finding new lower bounds for the spectral radius of the graph that are sufficient for the traceability property. Moreover, we develop the recognition algorithm of the existence of a perfect matching in a graph. This algorithm uses the concept of a (κ,τ)-regular set, which becomes polynomial in the class of graphs with a fixed cyclomatic number.

Refined notions of parameterized enumeration kernels with applications to matching cut enumeration

An enumeration kernel as defined by Creignou et al. (2017) [11] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

Mixed metric dimension of graphs with edge disjoint cycles

In a connected graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicyclic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles, where for each cycle Ci of G we define a unicyclic subgraph Gi of G in which Ci is the only cycle. Applying the result for unicyclic graph to the subgraph Gi of every cycle Ci then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of the mixed metric dimension yield a simple sharp upper bound on the mixed metric dimension, and we conclude the paper conjecturing that the analogous bound holds for general graphs with prescribed cyclomatic number.

Extremal mixed metric dimension with respect to the cyclomatic number

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)∪E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that every graph G with cyclomatic number c(G) satisfies mdim(G)≤L1(G)+2c(G) where L1(G) is the number of leaves in G. It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree ≥3. In this paper we determine that for every Θ-graph G, the mixed metric dimension mdim(G) equals 3 or 4, with 4 being attained if and only if G is a balanced Θ-graph. Thus, for balanced Θ-graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality mdim(G)=L1(G)+2c(G) holds.

Upper Bounds for the Independence Polynomial of Graphs at -1

The independence polynomial of a graph G is $$I(G;x)=\sum _{k=0}^{\alpha (G)} s_{k}\cdot x^{k}$$ , where $$s_{k}$$ and $$\alpha (G)$$ denote the number of independent sets of cardinality k and the independence number of G, respectively. We say that a cycle is a $$\tilde{3}$$ -cycle if its length is divisible by 3, otherwise a non- $$\tilde{3}$$ -cycle. Define $$\phi (G)$$ to be the decycling number of G. Engstrom proved that $$|I(G;-1)| \le 2^{\phi (G)}$$ for any graph G. In this paper, we first prove that $$|I(G;-1)|\le 2^{\beta (G)}-\beta (G)$$ for graphs with non- $$\tilde{3}$$ -cycles, where $$\beta (G)$$ is the cyclomatic number of G. Infinitely many examples show that there do exist graphs satisfying $$\beta (G)=\phi (G)$$ and containing non- $$\tilde{3}$$ -cycles. In such a case, it improves the Engstrom’s result. Furthermore, $$|I(G;-1)|\le 2^{k}$$ provided that all cycles of G are pairwise disjoint, where k is the number of $$\tilde{3}$$ -cycles of G. This provides a new perspective on estimating the independence polynomial at -1 of many special graphs. In the case G contains no vertices of degree 1, $$|I(G;-1)|\le 2^{\beta (G)}-1$$ if $$\beta (G)\ge 2$$ , and $$|I(G;-1)|\le 2^{\beta (G)-1}$$ if G contains a non- $$\tilde{3}$$ -cycle.

On a Conjecture about Degree Deviation Measure of Graphs

Let $G$ be an $n-$vertex graph with $m$ vertices‎. ‎The degree deviation measure of $G$ is defined as‎ ‎$s(G)$ $=$ $sum_{vin V(G)}|deg_G(v)‎- ‎frac{2m}{n}|,$ where $n$ and $m$ are the number of vertices and edges of $G$‎, ‎respectively‎. ‎The aim of this paper is to prove the Conjecture 4.2 of [J‎. ‎A‎. ‎de Oliveira‎, ‎C‎. ‎S‎. ‎Oliveira‎, ‎C‎. ‎Justel and N‎. ‎M‎. ‎Maia de Abreu‎, ‎Measures of irregularity of graphs‎, Pesq‎. ‎Oper.‎, ‎33 (2013) 383--398]‎. ‎The degree deviation measure of chemical graphs under some conditions on the cyclomatic number is also computed‎.

Graphs with given cyclomatic number extremal relatively to vertex degree function index for convex functions

Graphs with given cyclomatic number extremal relatively to vertex degree function index for convex functions

Wiener Complexity versus the Eccentric Complexity

Let wG(u) be the sum of distances from u to all the other vertices of G. The Wiener complexity, CW(G), is the number of different values of wG(u) in G, and the eccentric complexity, Cec(G), is the number of different eccentricities in G. In this paper, we prove that for every integer c there are infinitely many graphs G such that CW(G)−Cec(G)=c. Moreover, we prove this statement using graphs with the smallest possible cyclomatic number. That is, if c≥0 we prove this statement using trees, and if c<0 we prove it using unicyclic graphs. Further, we prove that Cec(G)≤2CW(G)−1 if G is a unicyclic graph. In our proofs we use that the function wG(u) is convex on paths consisting of bridges. This property also promptly implies the already known bound for trees Cec(G)≤CW(G). Finally, we answer in positive an open question by finding infinitely many graphs G with diameter 3 such that Cec(G)<CW(G).

Main eigenvalues of a graph and its Hamiltonicity

The concept of (κ,τ)-regular vertex set appeared in 2004. It was proved that the existence of many classical combinatorial structures in a graph like perfect matchings, Hamiltonian cycles, effective dominating sets, etc., can be characterized by (κ,τ)-regular sets the definition whereof is equivalent to the determination of these classical combinatorial structures. On the other hand, the determination of (κ,τ)-regular sets is closely related to the properties of the main spectrum of a graph. This paper generalizes the well-known properties of (κ,κ)-regular sets of a graph to arbitrary (κ,τ)-regular sets of graphs with an emphasis on their connection with classical combinatorial structures. We also present a recognition algorithm for the Hamiltonicity of the graph that becomes polynomial in some classes of graphs, for example, in the class of graphs with a fixed cyclomatic number.

Spectral problem of fullerene vibrations

Small vibrations of a graph of fullerene (truncated icosahedron) is considered each edge of which is a so-called Stieltjes string (a massless thread bearing finite number of point masses) symmetric with respect to its midpoint. The spectral problem is obtained by imposing the continuity and balance of forces conditions at the vertices. It is shown that when all the edges of the graph are the same then due to the symmetry of the problem there are multiple eigenvalues. The maximal multiplicity of an eigenvalue of such problem is 32, exactly the value which is maximal for cyclically connected graphs, i.e. $\mu+1$ where $\mu$ is the cyclomatic number of the graph.

Bounds on metric dimensions of graphs with edge disjoint cycles

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V(G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it distinguishes E(G). In this paper these invariants are considered first for unicyclic graphs, and it is shown that the vertex and edge metric dimensions obtain values from two particular consecutive integers, which can be determined from the structure of the graph. In particular, as a consequence, we obtain that these two invariants can differ by at most one for a same unicyclic graph. Next we extend the results to graphs with edge disjoint cycles (i.e. cactus graphs) showing that the two invariants can differ by at most c, where c is the number of cycles in such a graph. We conclude the paper with a conjecture that generalizes the previously mentioned consequences to graphs with prescribed cyclomatic number c by claiming that the difference of the invariant is still bounded by c.

Indicators for assessing the robustness of metapopulations against habitat loss

Habitat loss and fragmentation resulting from environmental changes are main drivers of global biodiversity loss, as the survival of metapopulations relies on the ability of individuals to disperse among suitable habitat patches. To prioritize conservation efforts, methods are needed for evaluating the robustness of metapopulations against habitat loss. We therefore investigate this robustness for different degrees of habitat loss, for different types of habitat loss (random, peripheral, and contagious) and of habitat networks, and for species differing in their local-extinction risks and dispersal ranges. In particular, we analyse several standard network types (with random, regular, small-world, or scale-free structure) and compare them with several alternative network types derived from real-world two-dimensional habitat landscapes (with random, clustered, or contiguous habitat allocation). Furthermore, we investigate how well 29 different graph-theoretic metrics of habitat networks can serve as indicators of metapopulation robustness against habitat loss – as this approach, where feasible, allows replacing complex simulation-based predictions with simple indicator-based predictions. We find that responses of species to habitat loss on the considered landscape-based habitat networks qualitatively differ from those on the considered standard habitat networks. This suggests that results obtained for the latter, albeit widely examined in the literature, can be unrepresentative and misleading. As expected, species with high risks of local extinction and short dispersal ranges are particularly vulnerable to habitat loss, across all considered types of habitat loss and habitat networks. The graph-theoretic network metric that best explains the robustness of metapopulations against habitat loss depends on the considered types of species, habitat networks, and habitat loss. None of the examined metrics give consistently reliable predictions under all circumstances. For sensitive species, characterized by high local-extinction risks and short dispersal ranges, a network’s average clique size, redundancy, average degree, connectance, clustering coefficient, and average closeness centrality are the best indicators of metapopulation robustness. For landscape-based habitat networks, a network’s average clique size, beta coefficient, clustering coefficient, redundancy, and cyclomatic number work best. For contagious habitat loss, the network type has a particularly strong impact on the species-specific robustness against habitat loss. In summary, our study introduces a method for evaluating the robustness of metapopulations against habitat loss and shows that a network’s clustering coefficient, under a wide range of circumstances, is a particularly reliable indicator of this robustness.