Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

Copyright q 2012 X. Pai and S. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let Φ G, λ det λIn − L G ∑n k 0 −1 ck G λn−k be the characteristic polynomial of the Laplacian matrix of a graph G of order n. In this paper, we g...

Let [Formula: see text] be a connected graph on the vertex set text]. Then In this paper, we prove that if is unicyclic graph, then depth of bounded below by Also, characterize with and We compute one distinguished extremal Betti numbers If obtained attaching whiskers at some vertices cycle length text], show Furthermore, each these cases, classify uniqueness number graphs.

A team of r revolutionaries and a team of s spies play a game on a graph G. Initially, revolutionaries and then spies take positions at vertices. In each subsequent round, each revolutionary may move to an adjacent vertex or not move, and then each spy has the same option. The revolutionaries want to hold an unguarded meeting, meaning m revolutionaries at some vertex having no spy at the end of...

We introduce the concept of minimum edge cover for an induced subgraph in a graph. Let [Formula: see text] be unicyclic graph with unique odd cycle and its ideal. compute exact values all symbolic defects using describe one method to find quasi-polynomial associated ideal text]. classify class graphs when some power maximal annihilates any fixed Also those graphs, we Hilbert function module

Abstract The graph-parameters mainly studies vertex degree and distance between unordered vertices. There are many graph-parameters, this paper focuses on the Gutman index. For sake of discussion, puts forward concept “ contribution” based relationship index graph vertex-Gutman On basis, we discussed unicyclic with pendent edges in odd even parts, gave corresponding extremal graphs, which lays ...

For a graph, the SK index is equal to half of sum degrees vertices, SK1 product and SK2 square vertices. This paper shows simple unified approach greatest indices for unicyclic graphs by using some transformations characterizes these with first, second, third having order r ≥ 5 girth g 3, where length shortest cycle in graph.

A signless Laplacian eigenvalue of a graph G is called a main signless Laplacian eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this paper, some necessary and sufficient conditions for a graph with one main signless Laplacian eigenvalue or two main signless Laplacian eigenvalues are given. And the trees and unicyclic graphs with exactly two main signless L...

An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero, and it is well known that a graph has exactly one main eigenvalue if and only if it is regular. In this work, all connected unicyclic graphs with exactly two main eigenvalues are determined. c © 2006 Elsevier Ltd. All rights reserved.