We show that attaching a whisker (or pendant) at the vertices of cycle cover graph results in new with following property: all symbolic powers its ideal are Koszul or, equivalently, componentwise linear. This extends previous work where whiskers were added to or vertex graph.

Let G be a graph with set of vertices V(G)(|V(G)|=n) and edge E(G). Very recently, new degree-based molecular structure descriptor, called Sombor index is denoted by SO(G) defined as SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) the degree vertex vi in G. In this paper we present some lower upper bounds on terms parameters (clique number, chromatic number pendant vertices, etc.) characterize...

The domination graph of a digraph has the same vertices as the digraph with an edge between two vertices if every other vertex loses to at least one of the two. Previously, the authors showed that the domination graph of a tournament is either an odd cycle with or without isolated and/or pendant vertices, or a forest of caterpillars. They also showed that any graph consisting of an odd cycle wi...

For a tree T , let iT (t) be the number of independent sets of size t in T . It is an open question, raised by Alavi, Malde, Schwenk and Erdős, whether the sequence (iT (t))t≥0 is always unimodal. Here we answer the question in the affirmative for some recursively defined families of trees, specifically paths with auxiliary trees dropped from the vertices in a periodic manner. In particular, ex...

The present study investigates the pendant number of certain graph classes; complement, line graphs, and total graphs. is minimum end vertices paths in a path decomposition graph. A it into subgraphs; i.e., sequence subset such that endpoints each edge appear one subsets vertex appears an adjacent sub-sequence subsets.

Let Wn = K1 ∨ Cn−1 be the wheel graph on n vertices, and let S(n, c, k) be the graph on n vertices obtained by attaching n− 2c− 2k − 1 pendant edges together with k hanging paths of length two at vertex v0, where v0 is the unique common vertex of c triangles. In this paper we show that S(n, c, k) (c > 1, k > 1) and Wn are determined by their signless Laplacian spectra, respectively. Moreover, w...

The connectivity index wα(G) of a graph G is the sum of the weights (d(u)d(v)) of all edges uv of G, where α is a real number (α 6= 0), and d(u) denotes the degree of the vertex u. Let T be a tree with n vertices and k pendant vertices. In this paper, we give sharp lower and upper bounds for w1(T ). Also, for −1 ≤ α < 0, we give a sharp lower bound and a upper bound for wα(T ).

Abstract In this article, we show that the rank of 2-Steiner distance matrix a caterpillar graph having N N vertices and p p pendant veritices is 2 − 1 2N-p-1 .

Let G be a simple connected graph and t be a given real number. The zero-order general Randić index αt(G) of G is defined as ∑ v∈V (G) d(v) t , where d(v) denotes the degree of v. In this paper, for any t , we characterize the graphs with the greatest and the smallest αt within two subclasses of connected unicyclic graphs on n vertices, namely, unicyclic graphs with k pendant vertices and unicy...