Pendant Vertices Research Articles

Minimum eccentric connectivity index for graphs with fixed order and fixed number of pendant vertices

The eccentric connectivity index of a connected graph G is the sum over all vertices v of the product dG(v)eG(v), where dG(v) is the degree of v in G and eG(v) is the maximum distance between v and any other vertex of G. We characterize, with a new elegant proof, those graphs which have the smallest eccentric connectivity index among all connected graphs of a given order n. Then, given two integers n and p with p ? n - 1, we characterize those graphs which have the smallest eccentric connectivity index among all connected graphs of order n with p pendant vertices.

On the Minimal General Sum-Connectivity Index of Connected Graphs Without Pendant Vertices

The general sum-connectivity index of a graph $G$ , denoted by $\chi _{_\alpha }(G)$ , is defined as $\sum _{uv\in E(G)}(d(u)+d(v))^{\alpha }$ , where $uv$ is the edge connecting the vertices $u,v\in V(G)$ , $d(w)$ denotes the degree of a vertex $w\in V(G)$ , and $\alpha $ is a non-zero real number. For $\alpha =-1/2$ and $n\geq 11$ , Wang et al. [On the sum-connectivity index, Filomat 25 (2011) 29–42] proved that $K_{2} + \overline {K}_{n-2}$ is the unique graph with minimum $\chi _{_\alpha }$ value among all the $n$ –vertex graphs having minimum degree at least 2, where $K_{2} + \overline {K}_{n-2}$ is the join of the 2-vertex complete graph $K_{2}$ and the edgeless graph $\overline {K}_{n-2}$ on $n-2$ vertices. Tomescu [2-connected graphs with minimum general sum-connectivity index, Discrete Appl. Math. 178 (2014) 135–141] proved that the result of Wang et al. holds also for $n\geq 3$ and $-1\leq \alpha . In this paper, it is shown that the aforementioned result of Wang et al. remains valid if the graphs under consideration are connected, $n\geq 6$ and $-1\leq \alpha , where $\alpha _{0}\approx -0.68119$ is the unique real root of the equation $\chi _{_\alpha }(K_{2} + \overline {K}_{4}) - \chi _{_\alpha }(C_{6})=0$ , and $C_{6}$ is the cycle on 6 vertices.

On Mostar index of trees with parameters

The Mostar index of a graph G is defined as the sum of absolute values of the differences between nu and nv over all edges uv of G, where nu and nv are respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u. We identify those trees with minimum and/or maximum Mostar index in the families of trees of order n with fixed parameters like the maximum degree, the diameter, number of pendent vertices by using graph transformations that decrease or increase the Mostar index.

Degree resistance distance of trees with some given parameters

The degree resistance distance of a graph $G$ is defined as $D_R(G)=sum_{i<j}(d(v_i)+d(v_j))R(v_i,v_j)$, where $d(v_i)$ is the degree of the vertex $v_i$, and $R(v_i,v_j)$ is the resistance distance between the vertices $v_i$ and $v_j$. Here we characterize the extremal graphs with respect to degree resistance distance among trees with given diameter, number of pendent vertices, independence number, covering number, and maximum degree, respectively.

Zero forcing number of a graph in terms of the number of pendant vertices

The zero forcing number of a graph has recently become an interesting graph parameter studied in its own right since its introduction by the ‘AIM Minimum Rank – Special Graphs Work Group’. In this article, we are interested in bounding the zero forcing number of a graph by the number of pendant vertices. Let and be the number of pendant vertices and the cyclomatic number of G. If G is a connected graph with at least one edge that is not a cycle, then , the extremal graphs whose zero forcing number attain the upper bound are characterized. For a connected graph G with at least one edge that is not a path, we prove , where is the number of similar equivalency classes of the set of all pendant vertices of G and two pendant vertices are said to be similar if they are the terminal vertices of a common major vertex. The extremal graphs with zero forcing number are also characterized.

On the extremal graphs for general sum-connectivity index ([formula omitted]) with given cyclomatic number when [formula omitted

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 χα(G) and is defined as ∑uv∈E(G)(du+dv)α, where uv is the edge connecting the vertices u,v∈V(G), du is the degree of the vertex u and α is a non-zero real number. 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. In this paper, it is proved that the unique graph obtained from the star Sn by adding ν edge(s) between a fixed pendant vertex u and ν other pendant vertices, has the maximum χα value in the collection of all n-vertex connected graphs having cyclomatic number ν with the constraints ν=5, n≥6, α>1 or 6≤ν≤n−2, α≥2. It is also proved that only those graphs which consist of (only) vertices of degrees 2, 3, such that no two vertices of degree 3 are adjacent, have the minimum χα value among all n-vertex connected graphs (and also among all n-vertex connected molecular graphs) having cyclomatic number ν with the conditions ν≥3, n≥5(ν−1) and α>1.

On the largest matching roots of graphs with a given number of pendent vertices

In this note, we study the bounds of the largest matching roots of graphs with 0,1,2,3,4,n−2, and n−1 pendent vertices. We also determine the extremal graphs with respect to these bounds.

The least Q-eigenvalue with fixed domination number

Denote by Lg, l the lollipop graph obtained by attaching a pendant path P=vgvg+1⋯vg+l (l ≥ 1) to a cycle C=v1v2⋯vgv1 (g ≥ 3). A Fg,l−graph of order n≥g+1 is defined to be the graph obtained by attaching n−g−l pendent vertices to some of the nonpendant vertices of Lg, l in which each vertex other than vg+l−1 is attached at most one pendant vertex. A Fg,l∘-graph is a Fg,l−graph in which vg is attached with pendant vertex. Denote by qmin the leastQ−eigenvalue of a graph. In this paper, we proceed on considering the domination number, the least Q-eigenvalue of a graph as well as their relation. Further results obtained are as follows:(i)some results about the changing of the domination number under the structural perturbation of a graph are represented;(ii)among all nonbipartite unicyclic graphs of order n, with both domination number γ and girth g (g≤n−1), the minimum qmin attains at a Fg,l-graph for some l;(iii)among the nonbipartite graphs of order n and with given domination number which contain a Fg,l∘-graph as a subgraph, some lower bounds for qmin are represented;(iv)among the nonbipartite graphs of order n and with given domination number n2.

General multiplicative Zagreb indices of trees

We study trees of given order, given number of pendant vertices, segments and branching vertices. We present upper and lower bounds on the general multiplicative Zagreb indices for trees of given order and number of pendant vertices/segments/branching vertices. Bounds on the first and the second multiplicative Zagreb indices are corollaries of the general results. We also characterize trees having the largest and the smallest indices.

Graceful labeling for open superstar of complete bipartite graphs S2(t·Km,n)

A graceful labeling of graph G with m edges is a one-to-one function f from vertices of G to the set {0, 1, 2, · · ·, m} such that the induced function f* from edges of G to the set {1, 2, 3, · · ·, m}, which is defined by the difference of its incident vertices labels, is a bijection. An open superstar of complete bipartite graphs is an integration of complete bipartite graphs and superstar graphs. A superstar graph is a graph obtained by replacing each edge of star St by a path P3 of three vertices. An open superstar of complete bipartite graph, S2(t·Km,n), is a graph obtained by identifying each pendant vertex of superstar with exactly one vertex of a complete bipartite graph Km,n. In this paper, we have given a graceful labeling for the open superstar of complete bipartite graphs S2(t·Km,n) for all positive integers m, n, and odd positive integers t.

On the multiplicity of α as an eigenvalue of Aα(G) of graphs with pendant vertices

Let G be a simple undirected graph. Let 0≤α≤1. LetAα(G)=αD(G)+(1−α)A(G) where D(G) and A(G) are the diagonal matrix of the vertex degrees of G and the adjacency matrix of G, respectively. Let p(G)>0 and q(G) be the number of pendant vertices and quasi-pendant vertices of G, respectively. Let mG(α) be the multiplicity of α as an eigenvalue of Aα(G). It is proved thatmG(α)≥p(G)−q(G) with equality if each internal vertex is a quasi-pendant vertex. If there is at least one internal vertex which is not a quasi-pendant vertex, the equalitymG(α)=p(G)−q(G)+mN(α) is determined in which mN(α) is the multiplicity of α as an eigenvalue of the matrix N. This matrix is obtained from Aα(G) taking the entries corresponding to the internal vertices which are non quasi-pendant vertices. These results are applied to search for the multiplicity of α as an eigenvalue of Aα(G) when G is a path, a caterpillar, a circular caterpillar, a generalized Bethe tree or a Bethe tree. For the Bethe tree case, a simple formula for the nullity is given.

On maximum signless Laplacian Estrada index of graphs with given parameters II

The signless Laplacian Estrada index of a graph $G$ is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$ where $q_1, q_2, \ldots, q_n$ are the eigenvalues of the signless Laplacian matrix of G. Following the previous work in which we have identified the unique graphs with maximum signless Laplacian Estrada index with each of the given parameters, namely, number of cut edges, pendent vertices, (vertex) connectivity, and edge connectivity, in this paper we continue our characterization for two further parameters: diameter and number of cut vertices.

On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

The zeroth-order general Randić index (usually denoted by [Formula: see text]) and variable sum exdeg index (denoted by [Formula: see text]) of a graph [Formula: see text] are defined as [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is degree of the vertex [Formula: see text], [Formula: see text] is a positive real number different from 1 and [Formula: see text] is a real number other than [Formula: see text] and [Formula: see text]. A segment of a tree is a path [Formula: see text], whose terminal vertices are branching or/and pendent, and all non-terminal vertices (if exist) of [Formula: see text] have degree 2. For [Formula: see text], let [Formula: see text], [Formula: see text], [Formula: see text] be the collections of all [Formula: see text]-vertex trees having [Formula: see text] pendent vertices, [Formula: see text] segments, [Formula: see text] branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randić index and variable sum exdeg index are determined from the collections [Formula: see text], [Formula: see text], [Formula: see text]. The obtained extremal trees for the collection [Formula: see text] are also extremal trees for the collection of all [Formula: see text]-vertex trees having fixed number of vertices with degree 2 (because the number of segments of a tree [Formula: see text] can be determined from the number of vertices of [Formula: see text] having degree 2 and vice versa).

The maximum spectral radius of uniform hypergraphs with given number of pendant edges

ABSTRACTIn this paper, we introduce the operations of grafting an edge and subdividing an edge on hypergraphs, and consider how spectral radius of a hypergraph behaves by grafting an edge or subdividing an edge. As an application, we determine the unique hypergraphs with the maximum spectral radius among all the uniform supertrees and all the connected uniform unicyclic hypergraphs with given number of pendant edges, respectively. Moreover, we determine the unique uniform supertree which attains the maximum spectral radius among all the uniform supertrees with given number of pendant vertices.

The least eigenvalue of graphs whose complements have only two pendent vertices

Let G be a simple graph and A(G) be the adjacency matrix of G. The eigenvalues of A(G) are referred to as the eigenvalues of G. In this paper, we characterize the graphs with the minimal least eigenvalue among all graphs whose complements are connected and have only two pendent vertices.

The kite graph is determined by its adjacency spectrum

The kite graph is obtained by appending the complete graph with p vertices to a pendant vertex of the path graph with q vertices. In this paper, we prove that the kite graph is determined by its adjacency spectrum for all p and q.

The maximum spectral radius of k-uniform hypergraphs with r pendent vertices

ABSTRACTLet G be a k-uniform hypergraph. The spectral radius of G is defined as the maximum modulus of eigenvalues of G. In this paper, we characterize the hypergraph on n vertices with maximum spectral radius among all connected k-uniform hypergraphs with exactly r pendent vertices, where or , and establish lower and upper bounds for the maximum spectral radius of connected hypergraphs in terms of the number of pendent vertices when ; We also characterize the unique hypergraphs with maximum spectral radius among all connected k-uniform hypergraphs and supertrees with given number of pendent edges, respectively.

On the Maximum ABC Index of Graphs With Prescribed Size and Without Pendent Vertices

The atom-bond connectivity (ABC) index is one of the most actively studied degree-based graph invariants, which are found in a vast variety of chemical applications. For a simple graph $G$ , it is defined as $ABC(G)=\sum _{uv \in E(G)} ({({d(u)+d(v)-2})/({d(u) d(v)})})^{1/2}$ , where $d(v)$ denotes the degree of a vertex $v$ of $G$ . Recently in [17] graphs with $n$ vertices, $2n-4$ and $2n-3$ edges, and maximum $ABC$ index were characterized. Here, we consider the next, more complex case, and characterize the graphs with $n$ vertices, $2n-2$ edges, and maximum $ABC$ index.

Face Sum Divisor Cordial Graphs

In this paper, we investigate the face sum divisor cordial labeling of switching of any vertex in cycle Cn, switching of a pendent vertex in path Pn and S′(K1,n).

Note on an upper bound for sum of the Laplacian eigenvalues of a graph

For a simple graph G with n vertices and m edges having Laplacian eigenvalues μ1(G)≥μ2(G)≥⋯≥μn(G), let Sk(G) be the sum of k largest Laplacian eigenvalues of G. In this note, we prove that if G is a connected graph of order n≥2 with m edges having clique number ω and vertex covering number τ, thenSk(G)≤k(τ+1)+m−ω(ω−1)2, with equality if k≤ω−1 and G is the graph obtained by joining n−ω pendant vertices with one of the vertices in Kω. Our work improves a recent work of Ganie et al.