Pendant Vertices Research Articles

On the Dα-spectral radius of cacti and bicyclic graphs

The Dα-matrix of a connected graph G is defined asDα(G)=αTr(G)+(1−α)D(G), where 0≤α<1, Tr(G) is the diagonal matrix of the vertex transmissions of G and D(G) is the distance matrix of G. The largest eigenvalue of Dα(G) is called the Dα-spectral radius of G. In this paper, we determine the unique graph with minimum Dα-spectral radius among the cacti of order n with k (k≥1) cycles and at least one pendent vertex. We also show that the Dα-spectral radius of Bn⁎ (the graph obtained from a star of order n by adding two nonadjacent edges) is less than the Dα-spectral radius of the most of bicyclic graphs of order n.

Extremal symmetric division deg index of molecular trees and molecular graphs with fixed number of pendant vertices

In 2018, Furtula et al. proved that the symmetric division deg index is a viable and applicable topological index in QSPR/QSAR investigations. In this article, we identify the extremal trees with respect to symmetric division deg index among all molecular trees with fixed number of pendant vertices. In addition, we get a lower bound on symmetric division deg index for all molecular (n,m,p)-graphs (n-order molecular graphs with m≥n edges and p>0 pendant vertices).

On Distance Irregular Labeling of Disconnected Graphs

A distance irregular k-labeling of a graph G is a function f : V (G) → {1, 2, . . . , k} such that the weights of all vertices are distinct. The weight of a vertex v, denoted by wt(v), is the sum of labels of all vertices adjacent to v (distance 1 from v), that is, wt(v) = P u∈N(v) f(u). If the graph G admits a distance irregular labeling then G is called a distance irregular graph. The distance irregularity strength of G is the minimum k for which G has a distance irregular k-labeling and is denoted by dis(G). In this paper, we derive a new lower bound of distance irregularity strength for graphs with t pendant vertices. We also determine the distance irregularity strength of some families of disconnected graphs namely disjoint union of paths, suns, helms and friendships.

On Trees with Greatest F-Invariant Using Edge Swapping Operations.

The F-index of a graph Q is defined as F(Q)=∑t∈V(Q)(dt)3. In this paper, we use edge swapping transformations to find the extremal value of the F-index among the class of trees with given order, pendent vertices, and diameter. We determine the trees with given order, pendent vertices, and diameter having the greatest F-index value. Also, the first five maximum values of F index among the class of trees with given diameter are determined.

On Spectral Characterization of Two Classes of Unicycle Graphs

Let G be a graph with n vertices, let A(G) be an adjacency matrix of G and let PA(G,λ) be the characteristic polynomial of A(G). The adjacency spectrum of G consists of eigenvalues of A(G). A graph G is said to be determined by its adjacency spectrum (DS for short) if other graphs with the same adjacency spectrum as G are isomorphic to G. In this paper, we investigate the spectral characterization of unicycle graphs with only two vertices of degree three. We use G21(s1,s2) to denote the graph obtained from Q(s1,s2) by identifying its pendant vertex and the vertex of degree two of P3, where Q(s1,s2) is the graph obtained by identifying a vertex of Cs1 and a pendant vertex of Ps2. We use G31(t1,t2) to denote the graph obtained from circle with the vertices v0v1⋯vt1+t2+1 by adding one pendant edge at vertices v0 and vt1+1, respectively. It is shown that G21(s1,s2) (s1≠4,6, s1≥3, s2≥3) and G31(t1,t2) (t1+t2≠2, t2≥t1≥1) are determined by their adjacency spectrum.

On Bond Incident Degree Indices of Connected Graphs with Fixed Order and Number of Pendent Vertices

The graphs having the maximum value of certain bond incident degree indices (including the second Zagreb index, general sum-connectivity index, and general zeroth-order Randić index) in the class of all connected graphs with fixed order and number of pendent vertices are characterized in this paper. The problem of finding graphs having the minimum values of the second Zagreb index and general zeroth-order Randić index from the aforementioned class of connected graphs is also addressed. One of the obtained results about the first Zagreb index has already been proved in the papers [I. Gutman, M. Kamran Jamil, N. Akhter, Trans. Combin. 4 (2015) 43-48] and [M. Enteshari, B. Taeri, MATCH Commun. Math. Comput. Chem. 86 (2021) 17-28]; however, it is proved here by another method with a short proof. Moreover, one of the obtained results concerning the second Zagreb index gives a partial solution to a problem attacked in the aforementiond paper of Enteshari and Taeri.

Graphs with eigenvalue −1 of multiplicity 2θ(G)+ρ(G)−1

Let G be a graph, m(G,λ) be the multiplicity of eigenvalue λ of A(G), θ(G) be the cyclomatic number of G, ρ(G) be the number of pendent vertices of G. Wang et al. proved that m(G,λ)≤2θ(G)+ρ(G)−1 for an arbitrary eigenvalue λ if G is not a cycle, and left an open problem to characterize the extremal graphs attaining the upper bound. We solve the open problem when λ=−1, that is a graph G satisfies m(G,−1)=2θ(G)+ρ(G)−1 if and only if it is obtained from a tree T with m(T,−1)=2θ(T)+ρ(T)−1 by attaching θ(G) cycles of order a multiple of 3 to θ(G) quasi-pendent vertices of T and delete related θ(G) pendent vertices, where ρ(T)≥θ(G).

A tight relation between series-parallel graphs and bipartite distance hereditary graphs

Bandelt and Mulder’s structural characterization of bipartite distance hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pendant vertices or twins (i.e., vertices with the same neighborhood as an existing one). Dirac and Duffin’s structural characterization of 2–connected series–parallel graphs asserts that such graphs can be built inductively starting from a single edge by adding either edges in series or in parallel. In this paper we give an elementary proof that the two constructions are the same construction when bipartite graphs are viewed as the fundamental graphs of a graphic matroid. We then apply the result to re-prove known results concerning bipartite distance hereditary graphs and series–parallel graphs and to provide a new class of polynomially-solvable instances for the integer multi-commodity flow of maximum value.

On eigenvalues and the energy of dendrimer trees

A dendrimer tree Dn,k is a rooted tree with root v0 in which the root vertex v0 has k children, the vertices u satisfying the distance d(v0,u)=i have k−1 children for 1≤i≤n−1, and the vertices u such that d(v0,u)=n are pendant vertices. In this paper, we obtain almost all of eigenvalues of Dn,k except n+1 eigenvalues which are just the roots of the matching polynomial of a weighted path with n+1 vertices v0,v1,…,vn and n edges (v0,v1),(v1,v2),(v2,v3)⋯,(vn−1,vn), of weights equal to k,k−1,k−1,⋯,k−1, and we obtain a formula to compute the energy of Dn,k.

English

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.

Graphs [formula omitted] with nullity [formula omitted

The nullity η(G) of a (simple) graph G is the multiplicity of 0 as an eigenvalue of the adjacency matrix of G. It was shown by Ma et al. (2016) that for a connected graph G with n(G) vertices, e(G) edges and p(G)(≥1) pendant vertices, η(G)≤2c(G)+p(G)−1, where c(G)=e(G)−n(G)+θ(G) is the cyclomatic number of G, θ(G) being the number of connected components of G. In this paper connected graphs G and trees T that satisfy respectively the conditions η(G)=2c(G)+p(G)−1 and η(T)=p(T)−1 are characterized. Besides, we identify the matched and mismatched vertices in a tree T that satisfies η(T)=p(T)−1 and characterize trees T that have a pendant vertex matched in T and satisfy η(T)=p(T)−2. We obtain new sharp lower and upper bounds for a tree T with p(T)≥3, given in terms of p(T) and the numbers of two special kinds of internal paths in T. It is also proved that for any integer c≥2, the set of values attained by η(G)−n(G)+2m(G), as G runs through all connected, leaf-free graphs G with cyclomatic number c, is {−c+1,−c+2,…,2c−3,2c−2}.

More on Sombor Index of Graphs

Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let G=(V(G),E(G)) be a graph. Then, the Sombor index of G is defined as SO(G)=∑uv∈E(G)dG2(u)+dG2(v). In this paper, we give some lemmas that can be used to compare the Sombor indices between two graphs. With these lemmas, we determine the graph with maximum SO among all cacti with n vertices and k cut edges. Furthermore, the unique graph with maximum SO among all cacti with n vertices and p pendant vertices is characterized. In addition, we find the extremal graphs with respect to SO among all quasi-unicyclic graphs.

Decomposition of cartesian product of complete graphs into sunlet graphs of order eight

For any integer $k\geq 3$, we define the sunlet graph of order $2k$, denoted by $L_{2k}$, as the graph consisting of a cycle of length $k$ together with $k$ pendant vertices such that, each pendant vertex adjacent to exactly one vertex of the cycle so that the degree of each vertex in the cycle is $3$. In this paper, we establish necessary and sufficient conditions for the existence of decomposition of the Cartesian product of complete graphs into sunlet graphs of order eight.

Multiplicative version of eccentric connectivity index

The eccentric connectivity index is defined for a connected graph G as the summation of the terms ɛG(u)+ɛG(v) over all edges uv of G, in which ɛG(u) denotes the eccentricity of the vertex u in G. In this paper, we study a multiplicative version of this graph invariant under the name multiplicative eccentric connectivity index. We present a sharp lower bound on the multiplicative eccentric connectivity index of n-vertex trees with fixed diameter and a sharp upper bound on the multiplicative eccentric connectivity index of n-vertex trees with given number of pendent vertices. Using these results, the trees on n vertices with the first three minimum and maximum values of the multiplicative eccentric connectivity index are characterized. Moreover, we present sharp bounds on the multiplicative eccentric connectivity index of connected graphs in terms of certain graph parameters including the order, size, radius, diameter, and number of universal vertices, and study the relationship between this invariant and several other vertex-eccentricity-based invariants. In particular, the minimal graphs with respect to the multiplicative eccentric connectivity index in the set of all connected graphs of given order, connected graphs of given size, and unicyclic and bicyclic graphs of given order are characterized and a Nordhaus–Gaddum-type result for this invariant is given.

Distance irregularity strength of graphs with pendant vertices

A vertex \(k\)-labeling \(\phi:V(G)\rightarrow\{1,2,\dots,k\}\) on a simple graph \(G\) is said to be a distance irregular vertex \(k\)-labeling of \(G\) if the weights of all vertices of \(G\) are pairwise distinct, where the weight of a vertex is the sum of labels of all vertices adjacent to that vertex in \(G\). The least integer \(k\) for which \(G\) has a distance irregular vertex \(k\)-labeling is called the distance irregularity strength of \(G\) and denoted by \(\mathrm{dis}(G)\). In this paper, we introduce a new lower bound of distance irregularity strength of graphs and provide its sharpness for some graphs with pendant vertices. Moreover, some properties on distance irregularity strength for trees are also discussed in this paper.

On the Minimum General Sum‐Connectivity of Trees of Fixed Order and Pendent Vertices

For a graph G, its general sum‐connectivity is usually denoted by χα(G) and is defined as the sum of the numbers over all edges uv of G, where dG(u), dG(v) represent degrees of the vertices u, v, respectively, and α is a real number. This paper addresses the problem of finding graphs possessing the minimum χα value over the class of all trees with a fixed order n and fixed number of pendent vertices n1 for α &gt; 1. This problem is solved here for the case when 4 ≤ n1 ≤ (n + 5)/3 and α &gt; 1, by deriving a lower bound on χα for trees in terms of their orders and number of pendent vertices.

Maximising the number of connected induced subgraphs of unicyclic graphs

Denote by $\mathcal{G}(n,c,g,k)$ the set of all connected graphs of order $n$, having $c$ cycles, girth $g$, and $k$ pendant vertices. In this paper, we give a partial characterisation of the structure of those graphs in $\mathcal{G}(n,c,g,k)$ maximising the number of connected induced subgraphs. For the special case where $c=1$, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the corresponding maximum number given the following parameters: (1) order, girth, and number of pendant vertices; (2) order and girth; (3) order.

On Ambarzumian type theorems for tree domains

It is known that the spectrum of the spectral Sturm-Liouville problem on an equilateral tree with (generalized) Neumann's conditions at all vertices uniquely determines the potentials on the edges in the unperturbed case, i.e. case of the zero potentials on the edges (Ambarzumian's theorem). This case is exceptional, and in general case (when the Dirichlet conditions are imposed at some of the pendant vertices) even two spectra of spectral problems do not determine uniquely the potentials on the edges. We consider the spectral Sturm-Liouville problem on an equilateral tree rooted at its pendant vertex with (generalized) Neumann conditions at all vertices except of the root and the Dirichlet condition at the root. In this case Ambarzumian's theorem can't be applied. We show that if the spectrum of this problem is unperturbed, the spectrum of the Neumann-Dirichlet problem on the root edge is also unperturbed and the spectra of the problems on the complimentary subtrees with (generalized) Neumann conditions at all vertices except the subtrees' roots and the Dirichlet condition at the subtrees' roots are unperturbed then the potential on each edge of the tree is 0 almost everywhere.

Ordering Acyclic Connected Structures of Trees Having Greatest Degree‐Based Invariants

Being building block of data sciences, link prediction plays a vital role in revealing the hidden mechanisms that lead the networking dynamics. Since many techniques depending in vertex similarity and edge features were put forward to rule out many well‐known link prediction challenges, many problems are still there just because of unique formulation characteristics of sparse networks. In this study, we applied some graph transformations and several inequalities to determine the greatest value of first and second Zagreb invariant, SK and SK1 invariants, for acyclic connected structures of given order, diameter, and pendant vertices. Also, we determined the corresponding extremal acyclic connected structures for these topological indices and provide an ordering (with 5 members) giving a sequence of acyclic connected structures having these indices from greatest in decreasing order.

Computing Fault‐Tolerant Metric Dimension of Connected Graphs

For a connected graph, the concept of metric dimension contributes an important role in computer networking and in the formation of chemical structures. Among the various types of the metric dimensions, the fault‐tolerant metric dimension has attained much more attention by the researchers in the last decade. In this study, the mixed fault‐tolerant dimension of rooted product of a graph with path graph with reference to a pendant vertex of path graph is determined. In general, the necessary and sufficient conditions for graphs of order at least 3 having mixed fault‐tolerant generators are established. Moreover, the mixed fault‐tolerant metric generator is determined for graphs having shortest cycle length at least 4.