Pendant Vertices Research Articles

On the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertex

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(G)) and Randić index (R(G)) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as ABC−R index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of ABC−R for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for ABC−R index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory.

On the general degree-eccentricity index of a graph

We define the general degree-eccentricity index of a connected graph G as $$DEI_{a,b}(G) = \sum _{v \in V(G)} d_{G}^{a}(v) ecc_{G}^{b}(v)$$ for $$a,b \in {\mathbb {R}}$$ , where V(G) is the vertex set of G, $$d_{G} (v)$$ is the degree of a vertex v and $$ecc_{G}(v)$$ is the eccentricity of v in G. Let $$I_1$$ be any index which decreases with the addition of edges and let $$I_2$$ be any index which increases with the addition of edges. We obtain sharp lower bounds on the $$I_1$$ index and the $$DEI_{a,b}$$ index, where $$a < 0$$ and $$b > 0$$ , and sharp upper bounds on the $$I_2$$ index and the $$DEI_{a,b}$$ index, where $$a > 0$$ and $$b < 0$$ , for connected graphs of given order in combination with given independence number, vertex cover number or minimum degree. We also present sharp upper bounds on the $$DEI_{a,b}$$ index, where $$a \ge 1$$ and $$b < 0$$ , for connected graphs of given order n in combination with given vertex connectivity, edge connectivity, number of pendant vertices, number of bridges or matching number $$\beta \le \frac{n}{4}$$ .

Binomial edge ideals of generalized block graphs

We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo–Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by [Formula: see text], where [Formula: see text] is the number of minimal cut sets of the graph [Formula: see text] and obtain an improved upper bound for the regularity in terms of the number of maximal cliques and pendant vertices of [Formula: see text].

On Variable Sum Exdeg Indices of Quasi-Tree Graphs and Unicyclic Graphs

In this work, by using the properties of the variable sum exdeg indices and analyzing the structure of the quasi-tree graphs and unicyclic graphs, the minimum and maximum variable sum exdeg indices of quasi-tree graphs and quasi-tree graphs with perfect matchings were presented; the minimum and maximum variable sum exdeg indices of unicyclic graphs with given pendant vertices and cycle length were determined.

On total vertex irregularity strength of generalized uniform cactus chain graphs with pendant vertices

In this research, we use the concept of a vertex irregular total k labeling and tvs(G). Specifically, we investigate tvs of generalized uniform cactus chain graphs with (n – 2)r pendant vertices and length r. The result is as follows: for n ≥ 6.

On the total edge irregularity strength of general uniform cactus chain graphs with pendant vertices

Let G(V, E) be a graph. Throughout this paper, we use the notions of edge irregular total k-labeling and total edge irregularity strength of G (tes (G)). We verify tes of general uniform cactus chain graphs having (n - 2)r pendant vertices and length r. The result obtained is as follows: for n ≥ 6.

The signed (| G |−1)-subdomination number of lollipop graphs

Let G be a lollipop graph obtained by appending a cycle Cn to a pendent vertex of a path Pm . In this paper, the signed (| G |−1) - subdomination number of a lollipop graph G is determined by classified discussion and exhaustived method.

On Kemeny's constant for trees with fixed order and diameter

ABSTRACT Kemeny's constant of a connected graph G is a measure of the expected transit time for the random walk associated with G. In the current work, we consider the case when G is a tree and, in this setting, we provide lower and upper bounds for in terms of the order n and diameter δ of G by using two different techniques. The lower bound is given as Kemeny's constant of a particular caterpillar tree and, as a consequence, it is sharp. The upper bound is found via induction, by repeatedly removing pendent vertices from G. By considering a specific family of trees – the broom-stars – we show that the upper bound is asymptotically sharp.

The maximum [formula omitted]-spectral radius and the majorization theorem of [formula omitted]-uniform supertrees

Let α be a real number with 0≤α<1, G be a uniform hypergraph, and Aα(G)=αD(G)+(1−α)A(G), where D(G) and A(G) are the diagonal degree tensor and the adjacency tensor of G, respectively. The spectral radius of Aα(G) is called the α-spectral radius of G. In this paper, some properties for the α-spectral radius of connected k-uniform hypergraphs are investigated, the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with a given degree sequence is characterized, and the majorization theorem for k-uniform supertrees is obtained. By applying the majorization theorem, we determine the k-uniform supertrees obtaining the three largest α-spectral radius among all k-uniform supertrees with n vertices, give a new proof ordering the eight largest spectral radius among all k-uniform supertrees with n vertices, propose an open problem for further research, and characterize the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with given different parameters, e.g., the maximum degree △, or the number of pendent vertices.

Cacti with maximal general sum-connectivity index

Let V(G) and E(G) be, respectively, the vertex set and edge set of a graph G. The general sum-connectivity index of a graph G is denoted by $$\chi _\alpha (G)$$ and is defined as $$\sum \limits _{uv\in E(G)}(d_u+d_v)^\alpha $$ , where uv is an edge that connect the vertices $$u,v\in V(G)$$ , $$d_u$$ is the degree of a vertex u and $$\alpha $$ is any non-zero real number. A cactus is a graph in which any two cycles have at most one common vertex. Let $$\mathscr {C}_{n,t}$$ denote the class of all cacti with order n and t pendant vertices. In this paper, a maximum general sum-connectivity index ( $$\chi _\alpha (G)$$ , $$\alpha >1$$ ) of a cacti graph with order n and t pendant vertices is considered. We determine the maximum general sum-connectivity index of n-vertex cacti graph. Based on our obtained results, we characterize the cactus with a perfect matching having the maximum general sum-connectivity index.

The least Laplacian eigenvalue of the unbalanced unicyclic signed graphs with $k$ pendant vertices

Let $\Gamma=(G,\sigma)$ be a signed graph and $L(\Gamma)=D(G)-A(\Gamma)$ be the Laplacian matrix of $\Gamma$, where $D(G)$ is the diagonal matrix of vertex degrees of the underlying graph $G$ and $A(\Gamma)$ is the adjacency matrix of $\Gamma$. It is well-known that the least Laplacian eigenvalue $\lambda_n$ is positive if and only if $\Gamma$ is unbalanced. In this paper, the unique signed graph (up to switching equivalence) which minimizes the least Laplacian eigenvalue among unbalanced connected signed unicyclic graphs with $n$ vertices and $k$ pendant vertices is characterized.

Reconstruction number of graphs with unique pendant vertex

A vertex-deleted subgraph of a graph G is called a card of G. The reconstruction number of G is the minimum number of cards of G that suffices to determine G uniquely. A P-graph (Yongzhi, 1988) is a connected graph of order p with exactly two blocks, only one of them is K2 and the other block has a vertex of degree p−2 other than the cut vertex. It is shown that the reconstruction number of P-graphs is three in most of the cases, which strengthens the result of Bollobas (1990). Finally, we conclude that the reconstruction number of most of the separable graphs (connected graphs with a cut vertex) with pendant vertices is three if the reconstruction number of all other connected graphs without pendant vertices is three.

Recovering the Shape of a Quantum Graph

Sturm–Liouville problems on simple connected equilateral graphs of $$\le 5$$ vertices and trees of $$\le 8$$ vertices are considered with Kirchhoff’s and continuity conditions at the interior vertices and Neumann conditions at the pendant vertices and the same potential on the edges. It is proved that if the spectrum of such problem is unperturbed (such as in case of zero potential) then this spectrum uniquely determines the shape of the graph and the zero potential. This is a generalization of the ’geometric’ Ambarzumian’s theorem of Boman et al. (Integral Equ. Oper. Theory 90:40, 2018. https://doi.org/10.1007/s00020-018-2467-1 ).

Extremum Modified First Zagreb Connection Index ofn-Vertex Trees with Fixed Number of Pendent Vertices

The modified first Zagreb connection indexZC1∗is a graph invariant that appeared about fifty years ago within a study of molecular modeling, and after a long time, it has been revisited in two papers ((Ali and Trinajstić, 2018) and (Naji et al., 2017)) independently. For a graphG, this graph invariant is defined asZC1∗G=∑v∈VGdvτv, wheredvis the degree of the vertexvandτvis the connection number ofv(that is, the number of vertices having distance 2 fromv). In this paper, the graphs with maximum/minimumZC1∗value are characterized from the class of alln-vertex trees with fixed number of pendent vertices (that are the vertices of degree 1).

Distance spectral radius of trees with given number of segments

A segment of a tree is a path-subtree whose terminal vertices are branching or pendant vertices. We determine the structure of trees that maximize the distance spectral radius among trees with given order and number of segments, and among trees with given order and number of vertices of degree 2, respectively.

General eccentric connectivity index of trees and unicyclic graphs

We introduce the general eccentric connectivity index of a graph G, ECIα(G)=∑v∈V(G)eccG(v)dGα(v) for α∈R, where V(G) is the vertex set of G, eccG(v) is the eccentricity of a vertex v and dG(v) is the degree of v in G. We present lower and upper bounds on the general eccentric connectivity index for trees of given order, trees of given order and diameter, and trees of given order and number of pendant vertices. Then we give lower and upper bounds on the general eccentric connectivity index for unicyclic graphs of given order, and unicyclic graphs of given order and girth. The upper bounds for trees of given order and diameter, and trees of given order and number of pendant vertices hold for α>1. All the other bounds are valid for 0<α≤1 or 0<α<1. We present all the extremal graphs, which means that our bounds are best possible.

Two classes of non-Leech trees

Let T be a tree of order n . For any edge labeling f : E → {1,2,3,...} the weight of a path P is the sum of the labels of the edges of P and is denoted by w ( P ). If the weights of the n C 2 paths in T are exactly 1, 2,..., n C 2 , then f is called a Leech labeling and a tree which admits a Leech labeling is called a Leech tree. In this paper we determine all Leech trees having diameter three. We also prove that the tree obtained from the path P n =( v 1 , v 2 ,..., v n ) by attaching a pendent vertex at v n -1 is not a Leech tree for all n ≥ 4.

General Multiplicative Zagreb Indices of Graphs with a Small Number of Cycles

We present lower and upper bounds on the general multiplicative Zagreb indices for bicyclic graphs of a given order and number of pendant vertices. Then, we generalize our methods and obtain bounds for the general multiplicative Zagreb indices of tricyclic graphs, tetracyclic graphs and graphs of given order, size and number of pendant vertices. We show that all our bounds are sharp by presenting extremal graphs including graphs with symmetries. Bounds for the classical multiplicative Zagreb indices are special cases of our results.

On unicyclic graphs with given number of pendent vertices and minimal energy

Let G(n,p) denote the set of unicyclic graphs with n vertices and p≥1 pendent vertices. Hua and Wang (2007) [4] claim to have determined the graph with minimal energy in G(n,p) except for the case when n=p+5 which was left as an open problem. This case was later solved by Huo et al. (2010) [5]. In this paper, we provide a counter example to show that the main result of Hua and Wang (2007) is not true. We also give the correct form of the characterization of unicyclic graphs with given number of pendent vertices and minimal energy. We make use of the well known Coulson's integral formula for the energy of graphs. Our proof is more general and also includes the case n=p+5 considered by Huo, Ji and Li (2010).