In this discrete time process, guards are allowed to traverse edges of a network under the constraint that the neighbourhood of each vertex must be visited within a specified time interval t. Setting t = 0 the guards form a dominating set at all times. We show every rooted tree with relatively few leaves has a branch with a certain order and number of pendant vertices. This result is used to es...

Abstract. For a connected graph, the distance spectral radius is the largest eigenvalue of its distance matrix. Let U1 2k be the graph obtained from C3 by attaching a path of length n− 3 at one vertex. Let U2 2k be the graph obtained from C3 by attaching a pendant edge together with k − 2 paths of length 2 at the same vertex. In this paper, it is proved that U1 2k (resp., U 2 2k) is the unique ...

Given a graph G, an edge-coloring or G with colors 1,2.3.. is consecutive if the colors of edges incident to each vertex fomm an interval of integer-s. This papelis devoted to bipartite graphs which do not have such a coloring of edges. We investigate their consecutive coloring deficiency, or shortly the deficiency d(G) of G, i.e. the minimum number of pendant edges whose attachment to G makes ...

In this paper we find formulas for group inverses of Laplacians of weighted trees. We then develop a relationship between entries of the group inverse and various distance functions on trees. In particular, we show that the maximal and minimal entries on the diagonal of the group inverse correspond to certain pendant vertices of the tree and to a centroid of the tree, respectively. We also give...

LetGw be a weighted graph. The number of the positive, negative and zero eigenvalues in the spectrum of Gw are called positive inertia index, negative inertia index and nullity of Gw, and denoted by i+(Gw), i−(Gw), i0(Gw), respectively. In this paper, sharp lower bound on the positive (resp. negative) inertia index of weighted bicyclic graphs of order n with pendant vertices is obtained. Moreov...

We study state transfer in quantum walks on graphs relative to the adjacency matrix. Our motivation is to understand how the addition of pendant subgraphs affect state transfer. For two graphs G and H, the Frucht-Harary corona product G ◦H is obtained by taking |G| copies of the cone K1 + H and by identifying the conical vertices according to G. Our work explores conditions under which the coro...

The harmonic index H (G) of a graph G is defined as the sum of the weights 2 du+dv of all edges uv of G, where du denotes the degree of a vertex u in G. In this paper, we determine (i) the trees of order n and m pendant vertices with the second smallest harmonic index, (ii) the trees of order n and diameter r with the smallest and the second smallest harmonic indices, and (iii) the trees of ord...

We consider a boundary value problem generated by Sturm-Liouville equations on the edges of a star-shaped graph. Thereby a continuity condition and a condition depending on the spectral parameter is imposed at the interior vertex, corresponding to the case of damping in the problem of small transversal vibrations of a star graph of smooth inhomogeneous strings. At the pendant vertices Dirichlet...

A graph with n vertices is well-covered if every maximal independent set is a maximum independent set, and very well-covered if every maximal independent set has size n ⁄2. In this paper, we study these graphs from an algorithmic complexity point of view. We show that wellcovered graph recognition is co-NP-complete and that several other problems are NP-complete for well-covered graphs. A numbe...

A spectral problem occurring in description of small transverse vibrations a star graph Stieltjes strings is considered. At all but one pendant vertices Dirichlet conditions are imposed which mean that these clamped. One vertex (the root) can move with damping the direction orthogonal to equilibrium position strings. We describe spectrum such problem. The corresponding inverse lies recovering v...