Pendent Paths Research Articles

Effects on the algebraic connectivity of weighted graphs under edge rotations

For a weighted graph G, the rotation of an edge uv1 from v1 to a vertex v2 is defined as follows: delete the edge uv1, set w(uv2) as w(uv1)+w(uv2) if uv2 is an edge of G; otherwise, add a new edge uv2 and set w(uv2)=w(uv1), where w(uv1) and w(uv2) are the weights of the edges uv1 and uv2, respectively. In this paper, effects on the algebraic connectivity of weighted graphs under edge rotations are studied. For a weighted graph, a sufficient condition for an edge rotation to reduce its algebraic connectivity and a necessary condition for an edge rotation to improve its algebraic connectivity are proposed based on Fiedler vectors of the graph. As applications, we show that, by using a series of edge rotations, a pair of pendent paths (a pendent tree) of a weighted graph can be transformed into one pendent path (pendent edges attached at a common vertex) of the graph with the algebraic connectivity decreasing (increasing) monotonically. These results extend previous findings of reducing the algebraic connectivity of unweighted graphs by using edge rotations.

Extremal results on the Mostar index of trees with fixed parameters

For a graph [Formula: see text], the Mostar index of [Formula: see text] is the sum of [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the number of vertices of [Formula: see text] that have a smaller distance in [Formula: see text] to [Formula: see text] than to [Formula: see text], and analogously for [Formula: see text]. We determine all the graphs that maximize and minimize the Mostar index, respectively, over all trees in terms of some fixed parameters like the number of odd vertices, the number of vertices of degree two, and the number of pendent paths of fixed length.

Upper Bounds of AZI and ABC Index for Transformed Families of Graphs

Topological index is a mapping which corresponds underlying graph with a numeric value and invariant up to all the isomorphisms of graph. Our study is based on a partial open question regarding topological indices: for which members of n-vertex graph family, certain index has minimum or maximum value? In this work, we answered the above-mentioned question regarding AZI and ABC for transformed families of graphs Γ n k , l and A α Γ n k , l . We investigated the fact of pendent paths and the transformation A α over these indices and developed the tight upper bounds regarding these families of graphs. Moreover, we characterized transformed graphs associated with maximum values of these indices.

Families of graphs with twin pendent paths and the Braess edge

In the context of a random walk on an undirected graph, Kemeny's constant can measure the average travel time for a random walk between two randomly chosen vertices. We are interested in graphs that behave counter-intuitively in regard to Kemeny's constant: in particular, we examine graphs with a cut-vertex at which at least two branches are paths, regarding whether the insertion of a particular edge into a graph results in an increase of Kemeny's constant. We provide several tools for identifying such an edge in a family of graphs and for analysing asymptotic behaviour of the family regarding the tendency to have that edge; and classes of particular graphs are given as examples. Furthermore, asymptotic behaviours of families of trees are described.

An Approach to the Geometric-Arithmetic Index for Graphs under Transformations’ Fact over Pendent Paths

Graph theory is a dynamic tool for designing and modeling of an interconnection system by a graph. The vertices of such graph are processor nodes and edges are the connections between these processors nodes. The topology of a system decides its best use. Geometric-arithmetic index is one of the most studied graph invariant to characterize the topological aspects of underlying interconnection networks or graphs. Transformation over graph is also an important tool to define new network of their own choice in computer science. In this work, we discuss transformed family of graphs. Let Γ n k , l be the connected graph comprises by k number of pendent path attached with fully connected vertices of the n-vertex connected graph Γ . Let A α Γ n k , l and A α β Γ n k , l be the transformed graphs under the fact of transformations A α and A α β , 0 ≤ α ≤ l , 0 ≤ β ≤ k − 1 , respectively. In this work, we obtained new inequalities for the graph family Γ n k , l and transformed graphs A α Γ n k , l and A α β Γ n k , l which involve GA Γ . The presence of GA Γ makes the inequalities more general than all those which were previously defined for the GA index. Furthermore, we characterize extremal graphs which make the inequalities tight.

An Approach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of n -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.

Alpha graphs with different pendent paths

Graceful labelings are an effective tool to find cyclic decompositions of complete graphs and complete bipartite graphs. The strongest kind of graceful labeling, the α-labeling, is in the center of the research field of graph labelings, the existence of an α-labeling of a graph implies the existence of several, apparently non-related, other labelings for that graph. Furthermore, graphs with α-labelings can be combined to form new graphs that also admit this type of labeling. The standard way to combine these graphs is to identify every vertex of a base graph with a vertex of another graph. These methods have in common that all the graphs involved, except perhaps the base, have the same size. In this work, we do something different, we prove the existence of an α-labeling of a tree obtained by attaching paths of different lengths to the vertices of a base path, in such a way that the lengths of the pendent paths form an arithmetic sequence with difference one, where consecutive vertices of the base path are identified with paths which lengths are consecutive elements of the sequence. These α-trees are combined in several ways to generate new families of α-trees. We also prove that these trees can be used to create unicyclic graphs with an α-labeling. In addition, we show that the pendent paths can be substituted by equivalent α-trees to produce new α-trees, obtaining in this manner a quite robust category of α-trees.

The maximal \alpha-index of trees with k pendent vertices and its computation

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. The $\alpha-$ index of $G$ is the spectral radius $\rho_{\alpha}\left( G\right)$ of the matrix $A_{\alpha}\left( G\right)=\alpha D\left( G\right) +(1-\alpha)A\left( G\right)$ where $\alpha \in [0,1]$. Let $T_{n,k}$ be the tree of order $n$ and $k$ pendent vertices obtained from a star $K_{1,k}$ and $k$ pendent paths of almost equal lengths attached to different pendent vertices of $K_{1,k}$. It is shown that if $\alpha\in\left[ 0,1\right) $ and $T$ is a tree of order $n$ with $k$ pendent vertices then% \[ \rho_{\alpha}(T)\leq\rho_{\alpha}(T_{n,k}), \] with equality holding if and only if $T=T_{n,k}$. This result generalizes a theorem of Wu, Xiao and Hong \cite{WXH05} in which the result is proved for the adjacency matrix ($\alpha=0$). Let $q=[\frac{n-1}{k}]$ and $n-1=kq+r$, $0 \leq r \leq k-1$. It is also obtained that the spectrum of $A_{\alpha}(T_{n,k})$ is the union of the spectra of two special symmetric tridiagonal matrices of order $q$ and $q+1$ when $r=0$ or the union of the spectra of three special symmetric tridiagonal matrices of order $q$, $q+1$ and $2q+2$ when $r \neq 0$. Thus the $\alpha-$ index of $T_{n,k}$ can be computed as the largest eigenvalue of the special symmetric tridiagonal matrix of order $q+1$ if $r=0$ or order $2q+2$ if $r\neq 0$.

The Least Eigenvalue of the Graphs Whose Complements Are Connected and Have Pendent Paths

AbstractThe adjacency matrix of a graph is a matrix which represents adjacent relation between the vertices of the graph. Its minimum eigenvalue is defined as the least eigenvalue of the graph. Let Gnbe the set of the graphs of order n, whose complements are connected and have pendent paths. This paper investigates the least eigenvalue of the graphs and characterizes the unique graph which has the minimum least eigenvalue in Gn.

The minimal-ABC trees with B1-branches.

The atom-bond connectivity index (or, for short, ABC index) is a molecular structure descriptor bridging chemistry to graph theory. It is probably the most studied topological index among all numerical parameters of a graph that characterize its topology. For a given graph G = (V, E), the ABC index of G is defined as , where di denotes the degree of the vertex i, and ij is the edge incident to the vertices i and j. A combination of physicochemical and the ABC index properties are commonly used to foresee the bioactivity of different chemical composites. Additionally, the applicability of the ABC index in chemical thermodynamics and other areas of chemistry, such as in dendrimer nanostars, benzenoid systems, fluoranthene congeners, and phenylenes is well studied in the literature. While finding of the graphs with the greatest ABC-value is a straightforward assignment, the characterization of the tree(s) with minimal ABC index is a problem largely open and has recently given rise to numerous studies and conjectures. A B1-branch of a graph is a pendent path of order 2. In this paper, we provide an important step forward to the full characterization of these minimal trees. Namely, we show that a minimal-ABC tree contains neither 4 nor 3 B1-branches. The case when the number of B1-branches is 2 is also considered.

On the α-index of graphs with pendent paths

Let G be a graph with adjacency matrix A(G) and let D(G) be the diagonal matrix of the degrees of G. For every real α∈[0,1], write Aα(G) for the matrixAα(G)=αD(G)+(1−α)A(G). This paper presents some extremal results about the spectral radius ρα(G) of Aα(G) that generalize previous results about ρ0(G) and ρ1/2(G).In particular, write Bp,q,r be the graph obtained from a complete graph Kp by deleting an edge and attaching paths Pq and Pr to its ends. It is shown that if α∈[0,1) and G is a graph of order n and diameter at least k, thenρα(G)≤ρα(Bn−k+2,⌊k/2⌋,⌈k/2⌉), with equality holding if and only if G=Bn−k+2,⌊k/2⌋,⌈k/2⌉. This result generalizes results of Hansen and Stevanović [5], and Liu and Lu [7].

On the Trees With Maximal Augmented Zagreb Index

The augmented Zagreb index (AZI), a variant of the well-known atom-bond connectivity (ABC) index, was shown to have the best predicting ability for a variety of physicochemical properties among several tested vertex-degree-based topological indices. However, contrasting to the extensive research on Problem A: characterizing $n$ -vertex tree(s) with minimal ABC index, few works have been done on Problem B: characterizing $n$ -vertex tree(s) with maximal AZI. Ali and Bhatti conjectured that a tree has maximal AZI iff it has minimal ABC index, with the implication that Problem B is as difficult as Problem A. In this paper, we first prove that among connected graphs with given degree sequence, there exists a breadth-first searching graph maximizing the AZI. Using this, an efficient algorithm based on tree degree sequences is designed to search the $n$ -vertex tree(s) with maximal AZI up to $n=200$ . We find that the balanced double star uniquely maximizes the AZI for $19\le n\le 200$ , and consequently, we disprove the aforementioned conjecture posed by Ali and Bhatti. Naturally, the balanced double star is conjectured to be the unique tree with maximal AZI among $n$ -vertex trees for $n\ge 19$ . Toward our conjecture, we prove that all the pendent paths are of length 1 in an $n$ -vertex tree with maximal AZI if $n\ge 19$ .

On structural properties of trees with minimal atom-bond connectivity index IV: Solving a conjecture about the pendent paths of length three

The atom-bond connectivity (ABC) index is one of the most investigated degree-based molecular structure descriptors with a variety of chemical applications. It is known that among all connected graphs, the trees minimize the ABC index. However, a full characterization of trees with a minimal ABC index is still an open problem. By now, one of the proved properties is that a tree with a minimal ABC index may have, at most, one pendent path of length 3, with the conjecture that it cannot be a case if the order of a tree is larger than 1178. Here, we provide an affirmative answer of a strengthened version of that conjecture, showing that a tree with minimal ABC index cannot contain a pendent path of length 3 if its order is larger than 415.

On structural properties of trees with minimal atom-bond connectivity index III: Trees with pendent paths of length three

The atom-bond connectivity (ABC) index is a degree-based graph topological index that found chemical applications, including those of predicting the stability of alkanes and the strain energy of cycloalkanes. Several structural properties of the trees with minimal ABC index were proved recently, however the complete characterization of the minimal-ABC trees is still an open problem. It is known that minimal-ABC trees can have at most one pendent path of length 3. It is also known that the minimal-ABC trees that have a pendent path of length 3 do not contain so-called Bk-branches, with k ≥ 4, and do not contain more than two B2-branches. Here, we improve the latter result by showing that minimal-ABC trees of order larger than 168 and with a pendent path of length 3 do not contain B2-branches. Moreover, we show that trees with minimal ABC index with a pendent path of length 3 do not contain B1-branches.

On a family of trees with minimal atom-bond connectivity index

Let G=(V,E) be a graph, di the degree of the vertex i, and ij the edge incident to the vertices i and j. The atom-bond connectivity index (or, simply, ABC index) is defined as ABC(G)=∑ij∈E(di+dj−2)/(didj). While this vertex-degree-based graph invariant is relatively well-known in chemistry, only recently a significant number of results emerged among the mathematical community. Though, several important problems remained open. One of them is the characterization of the tree(s) with minimal ABC index. In this paper, we will present some structural properties of one family of trees containing a pendent path of length 3 which would minimize the ABC index, mainly including: it contains no the so-called Bk with k⩾4, and contains at most two B2’s.

Spectral characterization of unicyclic graphs whose second largest eigenvalue does not exceed 1

Connected graphs with the same number of vertices and edges are called unicyclic graphs. All unicyclic graphs whose second largest eigenvalue does not exceed 1, denoted by U(λ2), are determined by Xu [10]. Let Gt,s denote the graph obtained by attaching t≥1 pendent paths of length 2 and s≥0 pendent edges to a vertex of a triangle. In this paper, we first prove that the graph Gt,s is determined by its adjacency spectrum. Then we prove that the graphs in U(λ2) are determined by their adjacency spectra except for four graphs.

An edge grafting theorem on the Estrada index of graphs and its applications

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. It can be used as an efficient measuring tool in a variety of fields. An edge grafting operation on a graph moves a pendent edge between two pendent paths. In this paper, we give an edge grafting theorem on the Estrada index of graphs. We also give some applications of this theorem.

On the Laplacian coefficients of trees with a perfect matching

Let T be a tree with n vertices and let ϕ ( T , λ ) = ∑ k = 0 n ( - 1 ) k c k ( T ) λ n - k be the characteristic polynomial of Laplacian matrix of T. It is well known that c n - 2 ( T ) is equal to the Wiener index of T, while c n - 3 ( T ) is equal to the modified hyper-Wiener index of T. For two n-vertex trees T 1 and T 2, write T 1 ⪯ T 2 if c k ( T 1 ) ⩽ c k ( T 2 ) for all 0 ⩽ k ⩽ n . Let Γ(n) be the set of all trees with 2 n vertices and perfect matchings except P 2n and A n-1,1, where P s is a path with s vertices and A s,t is a tree obtained from a star S t+1 with t+1 vertices by attaching s pendent paths of length 2 to the center of S t+1 . In this paper, we first give a graphic transformation that increases all Laplacian coefficients of an arbitrary tree, we then determine the minimum element and the second minimum element in Γ(n) under the partial order ⪯, and we finally identify the maximum element and second maximum element in Γ(n) under the partial order ⪯. Furthermore, we characterize some trees with extremal Wiener indices and Laplacian-like energies.

Some graft transformations and its application on a distance spectrum

Let D(G)=(di,j)n×n denote the distance matrix of a connected graph G with order n, where dij is equal to the distance between vi and vj in G. The largest eigenvalue of D(G) is called the distance spectral radius of graph G, denoted by ϱ(G). In this paper, we give some graft transformations that decrease and increase ϱ(G) and prove that the graph Sn′ (obtained from the star Sn on n (n is not equal to 4, 5) vertices by adding an edge connecting two pendent vertices) has minimal distance spectral radius among unicyclic graphs on n vertices; while Pn′ (obtained from a triangle K3 by attaching pendent path Pn−3 to one of its vertices) has maximal distance spectral radius among unicyclic graphs on n vertices.

On the Estrada and Laplacian Estrada indices of graphs

The Estrada index of a graph G is defined as EE ( G ) = ∑ i = 1 n e λ i , where λ 1 , λ 2 , … , λ n are the eigenvalues of G. The Laplacian Estrada index of a graph G is defined as LEE ( G ) = ∑ i = 1 n e μ i , where μ 1 , μ 2 , … , μ n are the Laplacian eigenvalues of G. An edge grafting operation on a graph moves a pendent edge between two pendent paths. We study the change of Estrada index of graph under edge grafting operation between two pendent paths at two adjacent vertices. As the application, we give the result on the change of Laplacian Estrada index of bipartite graph under edge grafting operation between two pendent paths at the same vertex. We also determine the unique tree with minimum Laplacian Estrada index among the set of trees with given maximum degree, and the unique trees with maximum Laplacian Estrada indices among the set of trees with given diameter, number of pendent vertices, matching number, independence number and domination number, respectively.