Pendant Vertices Research Articles

Edge perturbation on graphs with clusters: Adjacency, Laplacian and signless Laplacian eigenvalues

Let G be a simple undirected graph of order n. A cluster in G of order c and degree s, is a pair of vertex subsets (C,S), where C is a set of cardinality |C|=c≥2 of pairwise co-neighbor vertices sharing the same set S of s neighbors. Assuming that the graph G has k≥1 clusters (C1,S1),…,(Ck,Sk), consider a family of k graphs H1,…,Hk and the graph G(H1,…,Hk) which is obtained from G after adding the edges of the graphs H1,…,Hk whose vertex set of each Hj is identified with Cj, for j=1,…,k. The Laplacian eigenvalues of G(H1,…,Hk) remain the same, independently of the graphs H1,…,Hk, with the exception of |C1|+⋯+|Ck|−k of them. These new Laplacian eigenvalues are determined using a unified approach which can also be applied to the determination of a same number of adjacency and signless Laplacian eigenvalues when the graphs H1,…,Hk are regular. The Faria's lower bound on the multiplicity of the Laplacian eigenvalue 1 of a graph with pendant vertices is generalized. Furthermore, the algebraic connectivity and the Laplacian index of G(H1,…,Hk) remain the same, independently of the graphs H1,…,Hk.

Symmetry-adapted linear combinations for the eigenvalues and eigenvectors of reciprocal graphs

ABSTRACTThree classes of reciprocal graphs, viz. monocycle (GCn), linear chain (GLn) and star (GKn) with reciprocal pairs of eigenvalues (λ, 1/λ), are well known. Reciprocal graphs of monocycle (GCn) and linear chain (GLn) are obtained by putting a pendant vertex to each vertex of simple monocycle (Cn) and simple linear chain (Ln), respectively. A star graph of such kind is obtained by attaching a pendant vertex to the central vertex and to each of the (n − 1) peripheral vertices of the star graph (K1, (n−1)). An n-fold rotational axis of symmetry for GCn and (n − 1)-fold rotational axis of symmetry for GKn have been exploited for obtaining their respective condensed graphs. The condensed graph for GLn has been generated from that of GCn incorporating proper boundary conditions. Condensed graphs are lower dimensional graphs and are capable of keeping all eigeninformation in condensed form. Thus the eigensolutions (i.e. the eigenvalues and the eigenvectors) in analytical forms for such graphs are obtained by solving 2 × 2 or 4 × 4 determinants that in turn result in the charge densities and bond orders of the corresponding molecules in analytical forms. Some mathematical properties of the eigenvalues of such graphs have also been explored.

On the [formula omitted]-limited packing numbers in graphs

We give a sharp lower bound on the lower k-limited packing number of a general graph. Moreover, we establish a Nordhaus–Gaddum type bound on 2-limited packing number of a graph G of order n as L2(G)+L2(Ḡ)≤n+2. Also, we investigate the concepts of packing number (1-limited packing number) and open packing number in graphs with more details. In this way, by making use of the well-known result of Farber (1984) for strongly chordal graphs and its total version (2005) for trees we prove the new upper bound γ(G)≤(n−ℓ+δ′s)/(1+δ′) for every connected strongly chordal graph G of order n≥3 with ℓ pendant vertices and s support vertices, where δ′ is the minimum degree taken over all vertices that are not pendant vertices, and improve γt(T)≤(n+s)/2 for every tree T, that was first proved by Chellali and Haynes in 2004.

Decomposing certain equipartite graphs into sunlet graphs of length [formula omitted

For any integer r≥3, we define the sunlet graph of order 2r, denoted L2r, as the graph consisting of a cycle of length r together with r pendant vertices, each adjacent to exactly one vertex of the cycle. In this paper, we give necessary and sufficient conditions for decomposing the lexicographic product of the complete graph and the complete graph minus a 1-factor, with complement of the complete graph Km, (that is Kn⊗K̄m and Kn−I⊗K̄m, respectively) into sunlet graphs of order twice a prime.

Nullity of a graph in terms of the dimension of cycle space and the number of pendant vertices

Let G be a undirected graph without loops and multiple edges. By η(G),θ(G) and p(G) we respectively denote the nullity, the dimension of cycle space, and the number of pendant vertices of G. If each component of G contains at least two vertices, then it is proved that η(G)≤2θ(G)+p(G), the equality is attained if and only if every component of G is a cycle with size a multiple of 4.

The (signless) Laplacian spectral radii of c-cyclic graphs with n vertices, girth g and k pendant vertices

Let denote the class of c-cyclic graphs with n vertices, girth and pendant vertices. In this paper, we determine the unique extremal graph with largest signless Laplacian spectral radius and Laplacian spectral radius in the class of connected c-cyclic graphs with vertices, girth g and at most pendant vertices, respectively, and the unique extremal graph with largest signless Laplacian spectral radius of when and , and we also identify the unique extremal graph with largest Laplacian spectral radius in in the case and either and g is even or and g is odd. Our results extends the corresponding results of [Sci. Sin. Math. 2010;40:1017–1024, Electron. J. Combin. 2011; 18:p.183, Comput. Math. Appl. 2010;59:376–381, Electron. J. Linear Algebra. 2011;22:378–388 and J. Math. Res. Appl. 2014;34:379–391].

Maximum atom-bond connectivity index with given graph parameters

The atom-bond connectivity (ABC) index is a degree-based topological index. It was introduced due to its applications in modeling the properties of certain molecular structures and has been since extensively studied. In this note, we examine the influence on the extremal values of the ABC index by various graph parameters. More specifically, we consider the maximum ABC index of connected graphs of given order, with fixed independence number, number of pendent vertices, chromatic number and edge-connectivity respectively. We provide characterizations of extremal structures as well as some conjectures. Numerical analysis of the extremal values is also presented.

Ambarzumyan-type theorems on graphs with loops and double edges

In this work we consider inverse spectral problems for two types of graphs with loops and/or double edges. It is shown that analogs of Ambarzumyan's theorem are true for Sturm–Liouville problems on graphs with Neumann boundary conditions at the pendant vertices and Kirchhoff's conditions at the interior vertices.

On maximum signless Laplacian Estrada index of graphs with given parameters

For a simple graph G on n vertices, the signless Laplacian Estrada index is defined as S L E E ( G ) = ∑ i = 1 n e q i , where q 1 , q 2 , …, q n are the eigenvalues of the signless Laplacian matrix of G . In this paper, the unique graph on n vertices with maximum signless Laplacian Estrada index is determined among graphs with given number of cut edges, pendent vertices, (vertex) connectivity and edge connectivity, respectively.

On the rank of weighted graphs

Let be a weighted graph and be the adjacency matrix of . The rank of is the rank of . If the weight of each edge of is 1 or , is also called a signed graph. In this paper, we characterize weighted graphs with rank 2, weighted -free graphs with rank 3 and weighted graphs containing pendent vertices with rank 4. We also characterize signed graphs with rank 4.

The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices

The Wiener index is the sum of distances between all pairs of distinct vertices in a connected graph, which is the oldest topological index related to molecular branching. In this article, we give a condition to determine the graphs having the smallest Wiener index among all unicyclic graphs given number of pendant vertices, and we also determine the graphs having the smallest Wiener index among all unicyclic graphs given number of cut vertices.

Laplacian spectral characterization of dumbbell graphs and theta graphs

Let [Formula: see text] and [Formula: see text] denote the path and cycle on [Formula: see text] vertices, respectively. The dumbbell graph, denoted by [Formula: see text], is the graph obtained from two cycles [Formula: see text], [Formula: see text] and a path [Formula: see text] by identifying each pendant vertex of [Formula: see text] with a vertex of a cycle, respectively. The theta graph, denoted by [Formula: see text], is the graph formed by joining two given vertices via three disjoint paths [Formula: see text], [Formula: see text] and [Formula: see text], respectively. In this paper, we prove that all dumbbell graphs as well as all theta graphs are determined by their [Formula: see text]-spectra.

On the spectral characterization of kite graphs

The \textit{Kite graph}, denoted by $Kite_{p,q}$ is obtained by appending a complete graph $K_{p}$ to a pendant vertex of a path $P_{q}$. In this paper, firstly we show that no two non-isomorphic kite graphs are cospectral w.r.t the adjacency matrix. Let $G$ be a graph which is cospectral with $Kite_{p,q}$ and let $w(G)$ be the clique number of $G$. Then, it is shown that $w(G)\geq p-2q+1$. Also, we prove that $Kite_{p,2}$ graphs are determined by their adjacency spectrum.

Ordering non-bipartite unicyclic graphs with pendant vertices by the least Q-eigenvalue

A unicyclic graph is a connected graph whose number of edges is equal to the number of vertices. Fan et al. (Discrete Math. 313:903-909, 2013) and Liu et al. (Electron. J. Linear Algebra 26:333-344, 2013) determined, independently, the unique unicyclic graph whose least Q-eigenvalue attains the minimum among all non-bipartite unicyclic graphs of order n with k pendant vertices. In this paper, we extend their results and determine the first three non-bipartite unicyclic graphs of order n with k pendant vertices ordering by least Q-eigenvalue.

Matrices totally positive relative to a tree, II

If T is a labelled tree, a matrix A is totally positive relative to T, principal submatrices of A associated with deletion of pendent vertices of T are P-matrices, and A has positive determinant, then the smallest absolute eigenvalue of A is positive with multiplicity 1 and its eigenvector is signed according to T. This conclusion has been incorrectly conjectured under weaker hypotheses.

Solutions to unsolved problems on the minimal energies of two classes of trees

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph. Let Tn,p,Tn,d be the set of all trees of order n with p pendent vertices, diameter d, respectively. In this paper, we completely characterize the trees with second-minimal and third-minimal energy in Tn,p (Tn,d, respectively) for 4≤p≤n−9 (10≤d≤n−3, respectively), which solves the problems left in Ma (2014).

On the general sum-connectivity index of connected graphs with given order and girth

In this paper, we show that in the class of connected graphs $G$ of order $n\geq 3$ having girth at least equal to $k$, $3\leq k\leq n$, the unique graph $G$ having minimum general sum-connectivity index $\chi _{\alpha }(G)$ consists of $C_{k}$ and $n-k$ pendant vertices adjacent to a unique vertex of $C_{k}$, if $-1\leq \alpha <0$. This property does not hold for zeroth-order general Randi\' c index $^{0}R_{\alpha}(G)$.

Signed graphs with small positive index of inertia

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.

Characterization of graphs with given order, given size and given matching number that minimize nullity

Let G=(V(G),E(G)) be a finite undirected graph without loops and multiple edges, c(G)=|E(G)|−|V(G)|+θ(G) be the dimension of cycle spaces of G with θ(G) the number of connected components of G, m(G) be the matching number of G, and η(G) be the nullity of G. It was shown in Wang and Wong (2014) that the nullity η(G) of G is bounded by an upper bound and a lower bound as |V(G)|−2m(G)−c(G)≤η(G)≤|V(G)|−2m(G)+2c(G). Graphs with nullity attaining the upper bound have been characterized by Song et al. (2015). However, the problem of characterization of graphs whose nullity attain the lower bound is left open till now.In this paper, we are devoted to solve this open problem, proving that η(G)=|V(G)|−2m(G)−c(G) if and only if the cycles (if any) of G are pairwise vertex-disjoint and G can be switched, by a series of operations of deleting a pendant vertex together with its adjacent vertex, to an induced subgraph of G, which is the disjoint union of c(G) odd cycles and |V(G)|−2m(G)−c(G) isolated vertices.

The hyper-Wiener index of unicyclic graphs with n vertices and k pendent vertices

The hyper-Wiener index WW(G) is defined as with the summation going over all pairs of vertices in G. In this paper, we determine graphs with the minimum hyper-Wiener index among all the unicyclic graphs with n vertices and k pendent vertices.