Cyclomatic Number Research Articles

Inverse degree index of graphs with a given cyclomatic number

We investigate how the inverse degree index of graphs depends on their cyclomatic number. In particular, we provide sharp lower bounds on the inverse degree index over all graphs on a given number of vertices with a given cyclomatic number. We also deduce some structural properties of extremal graphs. Some open questions regarding the upper bound over the same class of graphs are discussed and some possible further developments are indicated.

Асимптотика числа помеченных планарных тетрациклических и пентациклических графов

A connected graph with a cyclomatic number k is said to be a k-cyclic graph. We obtain the formula for the number of labelled non-planar pentacyclic graphs with a given number of vertices, and find the asymptotics of the number of labelled connected planar tetracyclic and pentacyclic graphs with n vertices as n → ∞. We prove that under a uniform probability distribution on the set of graphs under consideration, the probability that the labelled tetracyclic graph is planar is asymptotically equal to 1089/1105, and the probability that the labeled pentacyclic graph is planar is asymptotically equal to 1591/1675.

Complete forcing numbers of graphs

The complete forcing number of a graph G with a perfect matching is the minimum cardinality of an edge set of G on which the restriction of each perfect matching M is a forcing set of M. This concept can be view as a strengthening of the concept of global forcing number of G. Došlić in 2007 has obtained that the global forcing number of a connected graph is at most its cyclomatic number. Motivated from this result, we obtain that the complete forcing number of a graph is no more than 2 times its cyclomatic number and characterize the matching covered graphs whose complete forcing numbers attain this upper bound and minus one, respectively. Besides, we present a method of constructing a complete forcing set of a graph. By using such method, we give closed formulas for the complete forcing numbers of wheels and cylinders.

Remarks on the Vertex and the Edge Metric Dimension of 2-Connected Graphs

The vertex (respectively edge) metric dimension of a graph G is the size of a smallest vertex set in G, which distinguishes all pairs of vertices (respectively edges) in G, and it is denoted by dim(G) (respectively edim(G)). The upper bounds dim(G)≤2c(G)−1 and edim(G)≤2c(G)−1, where c(G) denotes the cyclomatic number of G, were established to hold for cacti without leaves distinct from cycles, and moreover, all leafless cacti that attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves, and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper, we focus on Θ-graphs, as the most simple 2-connected graphs distinct from the cycle, and show that the the upper bound 2c(G)−1 holds for both metric dimensions of Θ-graphs; we characterize all Θ-graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2c(G)−1 in the class of leafless graphs besides already known extremal cacti and extremal Θ-graphs mentioned here.

The rank of a signed graph

The rank of a signed graph

Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two

Metric dimensions vs. cyclomatic number of graphs with minimum degree at least two

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

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

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

Graphs [formula omitted] with nullity [formula omitted

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

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

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

Mixed metric dimension of graphs with edge disjoint cycles

Extremal mixed metric dimension with respect to the cyclomatic number

Extremal mixed metric dimension with respect to the cyclomatic number

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.