Pendant Vertices Research Articles

Graphs with extremal incidence energy

For a simple connected graph G, the incidence energy IE(G) is defined as the sum of all singular values of its incidence matrix. In this paper, we characterize the graphs with the maximum incidence energies among all graphs with given chromatic number and given pendent vertex number, respectively. We also characterize the graphs with the minimum incidence energy among all graphs with given clique number. Especially, we characterize the tree with the minimum incidence energy among all trees with given pendent vertex number.

Modified Eccentric Connectivity of Generalized Thorn Graphs

The thorn graphGTof a given graphGis obtained by attachingt(>0)pendent vertices to each vertex ofG. The pendent edges, called thorns ofGT, can be treated asP2orK2, so that a thorn graph is generalized by replacingP2byPmandK2byKpand the respective generalizations are denoted byGPmandGKp. The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graphGTand the generalized thorn graphsGKpandGPm.

On the Eccentric Connectivity Index of Generalized Thorn Graphs

Let \(G\) be any connected graph of order \(p\), then the thorn graph \(G_T\) with code \((n_1,n_2,\ldots ,n_p)\) is obtained by adding \(n_i\) pendent vertices to each \(i\mathrm{th}\) vertex of \(G\). By treating the pendent edge of a thorn graph as \(P_2\), \(K_2\), \(K_{1,1}\), \(K_1 \circ K_1\) or \(P_1 \circ K_1\), the thorn graph is generalized by replacing \(P_2\) by \(P_m\), \(K_2\) by \(K_m\), \(K_{1,1}\) by \(K_{m,n}\), \(K_{1,1}\) by \(K_m \circ K_1\) and their respective generalized thorn graphs are denoted by \(G_P\), \(G_K, G_B\) and \(G_{KK}\). In this paper, we compute the eccentric connectivity index of these generalizations of thorn graphs.

Graph Homomorphisms between Trees

In this paper we study several problems concerning the number of homomorphisms of trees. We begin with an algorithm for the number of homomorphisms from a tree to any graph. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching generalization and a dual of Bollobás and Tyomkyn's result concerning the number of walks in trees.Some other main results of the paper are the following. Denote by $\hom(H,G)$ the number of homomorphisms from a graph $H$ to a graph $G$. For any tree $T_m$ on $m$ vertices we give a general lower bound for $\hom(T_m,G)$ by certain entropies of Markov chains defined on the graph $G$. As a particular case, we show that for any graph $G$, $$\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G),$$ where $\lambda$ is the largest eigenvalue of the adjacency matrix of $G$ and $H_{\lambda}(G)$ is a certain constant depending only on $G$ which we call the spectral entropy of $G$. We also show that if $T_m$ is any fixed tree and$$\hom(T_m,P_n)>\hom(T_m,T_n),$$for some tree $T_n$ on $n$ vertices, then $T_n$ must be the tree obtained from a path $P_{n-1}$ by attaching a pendant vertex to the second vertex of $P_{n-1}$.All the results together enable us to show that among all trees with fixed number of vertices, the path graph has the fewest number of endomorphisms while the star graph has the most.

On the general sum-connectivity index of connected unicyclic graphs with [formula omitted] pendant vertices

In this paper, we show that in the class of connected unicyclic graphs G of order n≥3 having 0≤k≤n−3 pendant vertices, the unique graph G having minimum general sum-connectivity index χα(G) consists of Cn−k and k pendant vertices adjacent to a unique vertex of Cn−k, if −1≤α<0. This property does not hold for zeroth-order general Randić index 0Rα(G).

Laplacian matrix of a weighted graph with new pendant vertices

The Laplacian matrix of a simple graph has been widely studied, as a consequence of its applications. However the Laplacian matrix of a weighted graph is still a challenge. In this work we provide the Moore-Penrose inverse of the Laplacian matrix of the graph obtained adding new pendant vertices to an initial graph, in terms of the Moore-Penrose inverse of the Laplacian matrix of the original graph. As an application we can compute the effective resistances and the Kirchhoff index of the new network.

The extremal problems on the inertia of weighted bicyclic graphs

Let Gw be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of Gw are called positive inertia index, negative inertia index and nullity of Gw, and denoted by i+(Gw), i-(Gw), i0(Gw), respectively. In this paper, sharp lower bound on the positive (respectively, negative) inertia index of weighted bicyclic graphs of order n with pendant vertices is obtained. Moreover, all the weighted bicyclic graphs of order n with at most two positive, two negative and at least n - 4 zero eigenvalues are identified, respectively.

Degree-based topological indices: Optimal trees with given number of pendents

A dynamic programming method is elaborated, enabling the characterization of trees with a given number of pendent vertices, for which a vertex-degree-based invariant (“topological index”) achieves its extremal value. The method is applied to the chemically interesting and earlier much studied such invariants: the first and second Zagreb index, and the atom–bond connectivity index.

English

Here we denote a diameter six tree by (a0; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr) , where a0 is the center of the tree; ai, i = 1, 2, . . . ,m , bj, j = 1, 2, . . . , n , and ck, k = 1, 2, . . . , r are the vertices of the tree adjacent to a0 ; each ai is the center of a diameter four tree, each bj is the center of a star, and each ck is a pendant vertex. Here we give graceful labelings to some new classes of diameter six trees (a0; a1, a2, . . . , am; b1, b2, . . . , bn; c1, c2, . . . , cr) in which we find diameter four trees consisting of four different combinations of odd, even, and pendant branches with the total number of branches odd. Here by a branch we mean a star, i.e. we call a star an odd branch if its center has an odd degree and an even branch if its center has an even degree.

First eigenvalue of nonsingular mixed graphs with given number of pendant vertices

Let G be a mixed graph and L(G) be the Laplacian matrix of the mixed graph G. The first eigenvalue of G is referred to the least nonzero eigenvalue of L(G). Let MG(n,k) be the set of nonsingular mixed graphs with n vertices and k pendant vertices, where n≥4. In this paper, up to a signature matrix, we determine the unique graph with the minimal first eigenvalue among all graphs in MG(n,k). Thus we obtain a lower bound for the first eigenvalue of a mixed graph in terms of the number of pendant vertices.

The Terminal Hosoya Polynomial of Some Families of Composite Graphs

Let G be a connected graph and let Ω(G) be the set of pendent vertices of G. The terminal Hosoya polynomial of G is defined as TH(G,t)∶=∑x,y∈Ω(G):x≠ytdG(x,y), where dG(x,y) denotes the distance between the pendent vertices x and y. In this note paper we obtain closed formulae for the terminal Hosoya polynomial of rooted product graphs and corona product graphs.

ALGEBRAIC CONNECTIVITY OF LOLLIPOP GRAPHS: A NEW APPROACH

The unicyclic graph Cn,gobtained by appending a cycle Cgof length g to a pendent vertex of a path on n - g vertices is the lollipop graph on n vertices. In [Algebraic connectivity of lollipop graphs, Linear Algebra Appl.434 (2011) 2204–2210], Guo et al. proved that a( Cn,g-1) &lt; a( Cn,g) for g ≥ 4, where a( Cn,g) is the algebraic connectivity of Cn,g. In this paper, we present a new approach which is quite different from that of Guo et al. in proving a( Cn,g-1) &lt; a( Cn,g) for g ≥ 4.

Some graphs determined by their (signless) Laplacian spectra

A graph G is L–DS (respectively, Q–DS) if there is no other non-isomorphic graph with the same (respectively, signless) Laplacian spectrum as G. Let G1∨G2 be the join graph of graphs G1 and G2, and Ur,n−r the graph obtained by attaching n−r pendent vertices to a vertex of Cr (the cycle of order r). In this paper, we prove that if G is L–DS and the algebraic connectivity of G is less than three, then Kt∨G is L–DS under certain condition, which extends the main result of Zhou and Bu (2012) [24]. Also, Ur,n−r is proved to be Q–DS for r⩾3.

Some Properties on the Harmonic Index of Molecular Trees

The harmonic index of a graph G is defined as the sum of weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u in G. In this paper, some general properties of the harmonic index for molecular trees are explored. Moreover, the smallest and largest values of harmonic index for molecular trees with given pendent vertices are provided, respectively.

On the largest eigenvalue of some bidegreed graphs

A simple connected non-regular graph is said to be -bidegreed, or biregular, if the vertices have degree from the set , with . We consider two classes of -bidegreed graphs denoted by and . A graph belongs to if: a) it is obtained from disjoint paths , , by identifying the vertices and the vertices , and the graph so obtained has n vertices; b) for each of the n vertices, pendant vertices are added, so that any vertex from any has degree . The class , , is similarly obtained by identifying all the vertices and from the ’s, into a single vertex. In this paper, we show that for any graph in or , the spectral radius of the adjacency matrix increases whenever the difference between the lengths of any two ’s increases. We also compute some bounds for the spectral radius when the lengths of the ’s tend to infinity. Finally, we discuss about bicyclic -bidegreed graphs with n degree vertices minimizing the spectral radius. We prove that in most cases such graphs do not belong to .

The combinatorial inverse eigenvalue problem II: all cases for small graphs

Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n by n matrices whose nonzero off-diagonal entries correspond to the edges of G. Given 2n - 1 real numbers \lambda_1\geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \cdots \geq \lambda_{n-1} \geq \mu_{n-1} \geq \lambda_{n-1}, and a vertex v of G, the question is addressed of whether or not there exists A in S(G) with eigenvalues \lambda_1, \ldots, \lambda_ n such that A(v) has eigenvalues \mu_1, \ldots, \mu_{n-1}, where A(v) denotes the matrix with vth row and column deleted. A complete solution can be given for the path on n vertices with v a pendant vertex and also for the star on n vertices with v the dominating vertex. The main result is a complete solution to this "\lambda, \mu" problem for all connected graphs on 4 vertices.

Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth

In this paper we consider the following problem: Over the class of all simple connected unicyclic graphs on n vertices with girth g (n, g being fixed), which graph minimizes the Laplacian spectral radius? Let Un,g be the lollipop graph obtained by appending a pendent vertex of a path on n − g (n > g) vertices to a vertex of a cycle on g ⩾ 3 vertices. We prove that the graph Un,g uniquely minimizes the Laplacian spectral radius for n ⩾ 2g − 1 when g is even and for n ⩾ 3g − 1 when g is odd.

Some relations on the ordering of trees by minimal energies between subclasses of trees

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. A tree is said to be non-starlike if it has at least two vertices with degree more than 2. A caterpillar is a tree in which a removal of all pendent vertices makes a path. Let $\mathcal{T}_{n,d}$ , $\mathbb{T}_{n,p}$ be the set of all trees of order n with diameter d, p pendent vertices respectively. In this paper, we investigate the relations on the ordering of trees and non-starlike trees by minimal energies between $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ . We first show that the first two trees (non-starlike trees, resp.) with minimal energies in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same for 3≤d≤n−2 (3≤d≤n−3, resp.). Then we obtain that the trees with third-minimal energy in $\mathcal{T}_{n,d}$ and $\mathbb{T}_{n,n-d+1}$ are the same when n≥11, 3≤d≤n−2 and d≠8; and the tree with third-minimal energy in $\mathcal{T}_{n,8}$ is the caterpillar with third-minimal energy in $\mathbb{T}_{n,n-7}$ for n≥11.

On the distance signless Laplacian spectral radius of graphs

The distance signless Laplacian spectral radius of a connected graph G is the spectral radius of the distance signless Laplacian matrix of G, defined as where is the diagonal matrix of vertex transmissions of G and is the distance matrix of G. In this paper, we determine the graphs with minimum distance signless Laplacian spectral radius among the trees, unicyclic graphs and bipartite graphs with fixed numbers of vertices, respectively, and determine the graphs with minimum distance signless Laplacian spectral radius among the connected graphs with fixed numbers of vertices and pendant vertices, and the connected graphs with fixed number of vertices and connectivity, respectively.

Further results on the distance spectral radius of graphs

Let D(G) be the distance matrix of a connected graph G. The distance spectral radius of G is the largest eigenvalue of D(G) and it has been proposed to be a molecular structure descriptor. In this article, we determine the unique trees with minimal and maximal distance spectral radii among trees with fixed bipartition. As a corollary, the trees with the first three minimal distance spectral radii are determined. Furthermore, we determine the unique trees with minimal distance spectral radii among n-vertex trees with fixed number of pendent vertices or fixed even diameter, respectively. We also propose a conjecture regarding the tree with minimal distance spectral radius among n-vertex trees with fixed odd diameter.