LARGEST EIGENVALUE OF A UNICYCLIC MIXED GRAPH

The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ 1(U)=n or λ 1(U)∈(n,n+1] are characterized.

On the ordering of the Kirchhoff indices of the complements of trees and unicyclic graphs

The Kirchhoff index Kf(G) of a graph G is defined to be the sum of the resistance distances between all pairs of vertices of G. In this paper, we develop a novel method for ordering the Kirchhoff indices of the complements of trees and unicyclic graphs. With this method, we determine the first five maximum values of Kf■ and the first four maximum values of Kf(ū),where ■ and ū are the complements of a tree T and unicyclic graph U, respectively.

Unicyclic graphs with extremal Lanzhou index

Very recently D.Vukicevic et al.[8]introduced a new topological index for a molecular graph G named Lanzhou index as∑_(u∈V(G))d_(u)d^(2)_(u),where d_(u)and d_(u)denote the degree of vertex u in G and in its complement respectively.Lanzhou index Lz(G)can be expressed as(n-1)M_(1)(G)-F(G),where M_(1)(G)and F(G)denote the first Zagreb index and the forgotten index of G respectively,and n is the number of vertices in G.It turns out that Lanzhou index outperforms M_(1)(G)and F(G)in predicting the logarithm of the octanol-water partition coefficient for octane and nonane isomers.It was shown that stars and balanced double stars are the minimal and maximal trees for Lanzhou index respectively.In this paper,we determine the unicyclic graphs and the unicyclic chemical graphs with the minimum and maximum Lanzhou indices separately.

A Note on the Nullity of Unicyclic Graphs

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. In this paper we show the expression of the nullity and nullity set of unicyclic graphs with n vertices and girth r, and characterize the unicyclic graphs with extremal nullity.

Ordering of Unicyclic Graphs with Perfect Matchings by Minimal Matching Energies

In 2012, Gutman and Wagner proposed the concept of the matching energy of a graph and pointed out that its chemical applications can go back to the 1970s. The matching energy of a graph is defined as the sum of the absolute values of the zeros of its matching polynomial. Let u and v be the non-isolated vertices of the graphs G and H with the same order, respectively. Let wi?be a non-isolated vertex of graph Gi?where i=1, 2, …, k. We use Gu(k)?(respectively, Hv(k)) to denote the graph which is the coalescence of G (respectively, H) and G1, G2,…, Gk?by identifying the vertices u (respectively, v) and w1, w2,…, wk. In this paper, we first present a new technique of directly comparing the matching energies of Gu(k)?and Hv(k), which can tackle some quasi-order incomparable problems. As the applications of the technique, then we can determine the unicyclic graphs with perfect matchings of order 2n with the first to the ninth smallest matching energies for all n≥211.

The lower bound of revised edge-Szeged index of unicyclic graphs with given diameter

Given a connected graph G,the revised edge-revised Szeged index is defined as Sz_(e)^(*)(G)=∑_(e=uv∈E_(G))(m_(u)(e)+m_(0)(e)/2)(m_(v)(e)+m_(0)(e)/w),where m_(u)(e),m_(v)(e)and m_(0)(e)are the number of edges of G lying closer to vertex u than to vertex u,the number of edges of G lying closer to vertex than to vertex u and the number of edges of G at the same distance to u and u,respectively.In this paper,by transformation and calculation,the lower bound of revised edge-Szeged index of unicyclic graphs with given diameter is obtained,and the extremal graph is depicted.

On the Spectral Moment of Quasi-Unicyclic Graphs

A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u0∈V(G) such that G-u0 is a unicyclic graph.Denote Q(n,d0)={G:G is a quasi-unicyclic graph of order n with G-u0 being a unicyclic graph and dG(u0)=d0}.Let A(G) be the adjacency matrix of a graph G,and let λ1(G),λ2(G),…,λn(G) be the eigenvalues in non-increasing order of A(G).The number n∑i=1λi^k(k=0,1,…,n-1) is called the k-th spectral moment of G,denoted by Sk(G).Let S(G)=(S0(G),S1(G),…,Sn-1(G)) be the sequence of spectral moments of G.For two graphs G1,G2,we have■ if for some k(k=1,2,…,n-1), and we have Si(G1)=Si(G2)(i=0,1,…,k-1) and Sk(G1)<Sk(G2).In this paper,we determine the second to the fourth largest quasi-unicyclic graphs,in an S-order,in the set Q(n,d0),respectively.

The Laplacian spectral radii of unicyclic and bicyclic graphs with n vertices and k pendant vertices

In this paper,we determine graphs with the largest Laplacian spectral radius among the unicyclic and the bicyclic graphs on n vertices with k pendant vertices,respectively.

Unicyclic Graphs with Nullity One

The nullity of a graph G is defined to be the multiplicity of the eigenvalue zero in its spectrum. In this paper we characterize the unicyclic graphs with nullity one in aspect of its graphical construction.

Unicyclic Graphs of Minimal Spectral Radius

It was conjectured by Li and Feng in 1979 that the unicyclic graph formed by a cycle of order g linking to an endvertex of a path of length k minimizes the spectral radius of all unicyclic graphs of order g ＋ k and girth g. In 1987, Cao proved that this conjecture is true for k ≥ g（g - 2）/8 and false for k = 2 and sufficiently large g. In this note, we show that g 〉12 suffices for the counterexample and give more counterexamples with large girth for any integer k 〉 1.

The Least Eigenvalue of Unicyclic Graphs with Application to Spectral Spread

Let U^(g)_(n)be the set of connected unicyclic graphs of order n and girth g.Let C(T_(1),T_(2),…,T_(g))∈U^(g)_(n)be obtained from a cycle v_(1)v_(2)…v_(g)v_(1)(in an anticlockwise direction)by identifying v_(i)with the root of a rooted tree T_(i)of order n_(i)for each i=1,2,…,g,where n_(i)≥1 and∑^(g)_(i=1)n_(i)=n.In this note,the graph with the minimal least eigenvalue(and the graph with maximal spread)in C(T_(1),T_(2),…,T_(g))is determined.

A Note on the Nullity of Unicyclic Graphs

The number of zero eigenvalues in the spectrum of the graph G is called its nullity and is denoted by η（G）. In this paper, we determine the all extremal unicyclic graphs achieving the fifth upper bound n - 6 and the sixth upperbound n - 7.

The Maximum α-spectral Radius of Unicyclic Hypergraphs with Fixed Diameter

For 0≤α<1,theα-spectral radius of an r-uniform hypergraph G is the spectral radius of A_(α)(G)=αD(G)+(1-α)A(G),where D(G)and A(G)are the diagonal tensor of degrees and adjacency tensor of G,respectively.In this paper,we show the perturbation ofα-spectral radius by contracting an edge.Then we determine the unique unicyclic hypergraph with the maximumα-spectral radius among all r-uniform unicyclic hypergraphs with fixed diameter.We also determine the unique unicyclic hypergraph with the maximumα-spectral radius among all r-uniform unicyclic hypergraphs with given number of pendant edges.

A Comparison between the Metric Dimension and Zero Forcing Number of Trees and Unicyclic Graphs

The metric dimension dim（G） of a graph G is the minimum number of vertices such that every vertex of G is uniquely determined by its vector of distances to the chosen vertices. The zero forcing number Z（G） of a graph G is the minimum eardinality of a set S of black vertices （whereas vertices in V（G）／S are colored white） such that V（G） is turned black after finitely many applications of ＂the color-change rule＂： a white vertex is converted black if it is the only white neighbor of a black vertex. We show that dim（T） ≤Z（T） for a tree T, and that dim（G）≤Z（G）＋I if G is a unicyclic graph; along the way, we characterize trees T attaining dim（T） = Z（T）. For a general graph G, we introduce the ＂cycle rank conjecture＂. We conclude with a proof of dim（T） - 2 ≤ dim（T ＋ e） ≤dim（T） ＋ 1 for e∈ E（T）.

The Linear Unicyclic Hypergraph with the Second or Third Largest Spectral Radius

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor of the hypergraph.It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph,and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph.In this paper we determine the unique linear unicyclic hypergraph with the second or third largest spectral radius,where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.

BICYCLIC GRAPHS WITH UNICYCLIC OR BICYCLIC INVERSES

A graph G is nonsingular if its adjacency matrix A(G)is nonsingular.A nonsingular graph G is said to have an inverse G+if A(G)-1 is signature similar to a nonnegative matrix.Let H be the class of connected bipartite graphs with unique perfect matchings.We present a characterization of bicyclic graphs in H which possess unicyclic or bicyclic inverses.

ON q-WIENER INDEX OF UNICYCLIC GRAPHS

The q-Wiener index of unicyclic graphs are determined in this work. As an example of its applications, an explicit expression of q-Wiener index of caterpillar cycles is presented.

A Note on Comparison Between the Wiener Index and the Zagreb Indices

In this note, we correct a wrong result in a paper of Das et al. with regard to the comparison between the Wiener index and the Zagreb indices for trees （Das K C, Jeon H, Trinajstic N. The comparison between the Wiener index and the Zagreb indices and the eccentric connectivity index for trees. Discrete Appl. Math., 2014, 171：35 41）, and give a simple way to compare the Wiener index and the Zagreb indices for trees. Moreover, the comparison between the Wiener index and the Zagreb indices for unicyclic graphs is carried out.

Optimisation-based Verification Process of Obstacle Avoidance Systems for Unicycle-like Mobile Robots

This paper presents an optimisatiombased verification process for obstacle avoidance systems of a unicycle-like mobile robot. It is a novel approach for the collision avoidance verification process. Local and global optimisation based verification processes are developed to find the worst-case parameters and the worst-case distance between the robot and an obstacle. The kinematic and dynamic model of the unicycle-like mobile robot is first introduced with force and torque as the inputs. The design of the control system is split into two parts. One is velocity and rotation using the robot dynamics, and the other is the incremental motion planning for robot kinematics. The artificial potential field method is chosen as a path planning and obstacle avoidance candidate technique for verification study as it is simple and widely used. Different optimisation algorithms are applied and compared for the purpose of verification. It is shown that even for a simple case study where only mass and inertia variations are considered, a local optimization based verification method may fail to identify the worst case. Two global optimisation methods have been investigated： genetic algorithms （GAs） and GLOBAL algorithms. Both of these methods successfully find the worst case. The verification process confirms that the obstacle avoidance algorithm functions correctly in the presence of all the possible parameter variations.

Review and Simulation of a Novel Formation Building Algorithm While Enabling Obstacle Avoidance

Technically, a group of more than two wheeled mobile robots working collectively towards a common goal are known as a multi-robot system. An increasing number of industries have implemented multi-robot systems to eliminate the risk of human injuries while working on hazardous tasks, and to improve productivity. Globally, engineers are continuously researching better, simple, and faster cooperative Control algorithms to provide a Control strategy where each agent in the robot formation can communicate effectively and achieve a consensus in their position, orientation and speed. This paper explores a novel Formation Building Algorithm and its global stability around a configuration vector. A simulation in MATLAB? was carried out to examine the performance of the Algorithm for two geometric formations and a fixed number of robots. In addition, an obstacle avoidance technique was presented assuming that all robots are equipped with range sensors. In particular, a uniform rounded obstacle is used to analyze the performance of the technique with the use of detailed geometric calculations.