Dumbbell Graphs Research Articles

Resistance distance of generalized wheel and dumbbell graph using symmetric {1}-inverse of Laplacian matrix

A new class of graphs called dumbbell graphs, denoted by DB(Wm,n) is the graph obtained from two copies of generalized wheel graph Wm,n, m ≥ 2, n ≥ 3. It is a graph on 2 (m + n) vertices obtained by connecting m-vertices in one copy with the corresponding vertices in the other copy. The resistance distance between two vertices vi and vj, denoted by rij , is defined as the effective electrical resistance between them if each edge of G is replaced by 1 ohm resistor. The Kirchhoff index is the sum of the resistance distances between all pairs of vertices in the graph. In this paper, we formulate the resistance distance of Wm,n and DB(Wm,n) using Symmetric {1}-inverse of Laplacian matrices. We provide examples to illustrate the proposed method and also obtain the Kirchhoff indices for these examples.

Dumbbell Graphs with Extremal (Reverse) Cover Cost

Dumbbell Graphs with Extremal (Reverse) Cover Cost

Regularity of bicyclic graphs and their powers

Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that [Formula: see text], for all [Formula: see text], when [Formula: see text] is a dumbbell graph with a connecting path having no more than two vertices.

On the minimum value of sum-Balaban index

We consider extremal values of sum-Balaban index among graphs on n vertices. We determine that the upper bound for the minimum value of the sum-Balaban index is at most 4.47934 when n goes to infinity. For small values of n we determine the extremal graphs and we observe that they are similar to dumbbell graphs, in most cases having one extra edge added to the corresponding extreme for the usual Balaban index. We show that in the class of balanced dumbbell graphs, those with clique sizes 2log(1+2)4n+o(n) have asymptotically the smallest value of sum-Balaban index. We pose several conjectures and problems regarding this topic.

On super edge-magic deficiency of volvox and dumbbell graphs

Let G=(V,E) be a finite, simple and undirected graph of order p and size q. A super edge-magic total labeling of a graph G is a bijection λ:V(G)∪E(G)→{1,2,…,p+q}, where the vertices are labeled with the numbers 1,2,…,p and there exists a constant t such that f(x)+f(xy)+f(y)=t, for every edge xy∈E(G). The super edge-magic deficiency of a graph G, denoted by μs(G), is the minimum nonnegative integer n such that G∪nK1 has a super edge-magic total labeling, or it is ∞ if there exists no such n.In this paper, we are dealing with the super edge-magic deficiency of volvox and dumbbell type graphs.

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.

A search for the minimum value of Balaban index

In this paper we consider graphs of order n with minimum Balaban index. Although the index was introduced 30 years ago, its minimum value and corresponding extremal graphs are still unknown, and it is unlikely that they can be precisely determined soon due to the mathematical intractability of the index. We show that this value is of order Θ(n−1). For small values of n we determine the extremal graphs and we observe that they are similar to dumbbell graphs. We find out that in the class of balanced dumbbell graphs those with clique sizes π/24n+o(n) and the path length n−o(n) have asymptotically the smallest value. We study dumbbell-like graphs in more detail, and we propose several conjectures regarding the structure of the extremal graphs.

SUPER STRONGLY PERFECTNESS OF SOME GRAPHS

A Graph G is Super Strongly Perfect Graph if every induced sub graph H of G possesses a minimal dominating set that meets all the maximal cliques of H. The structure of Super Strongly Perfect Graphs have been characterized by some classes of graphs like Cycle graphs, Circulant graphs, Complete graphs, Complete Bipartite graphs etc., In this paper, we have analysed some other graph classes like, Bicyclic graphs, Dumb bell graphs and Star graphs to characterize the structure of Super Strongly Perfect Graphs in a different way. By this we found the cardinality of minimal dominating set and maximal cliques of the above graphs. AMS Subject Classification: 05C75

Non-Isomorphic Dumbbell Graphs are Laplaican Cospectral

Non-Isomorphic Dumbbell Graphs are Laplaican Cospectral

Unordered multiplicity lists of wide double paths

Recently, Kim and Shader analyzed the multiplicities of the eigenvalues of a Φ -binary tree. We carry this discussion forward extending their results to a larger family of trees, namely, the wide double path, a tree consisting of two paths that are joined by another path. Some introductory considerations for dumbbell graphs are mentioned regarding the maximum multiplicity of the eigenvalues. Lastly, three research problems are formulated.

Spectral Characterizations of Dumbbell Graphs

A dumbbell graph, denoted by $D_{a,b,c}$, is a bicyclic graph consisting of two vertex-disjoint cycles $C_a$, $C_b$ and a path $P_{c+3}$ ($c \geq -1$) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of $D_{a,b,0}$ (without cycles $C_4$) with $\gcd(a,b)\geq 3$, and we complete the research started in [J.F. Wang et al., A note on the spectral characterization of dumbbell graphs, Linear Algebra Appl. 431 (2009) 1707–1714]. In particular we show that $D_{a,b,0}$ with $3 \leq \gcd(a,b) < a$ or $\gcd(a,b)=a$ and $b\neq 3a$ is determined by the spectrum. For $b=3a$, we determine the unique graph cospectral with $D_{a,3a,0}$. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell graphs.

WITHDRAWN: Spectral characterization of dumbbell graphs

WITHDRAWN: Spectral characterization of dumbbell graphs

A note on the spectral characterization of dumbbell graphs

The dumbbell graph, denoted by D a , b , c , is a bicyclic graph consisting of two vertex-disjoint cycles C a and C b joined by a path P c + 3 ( c ⩾ - 1 ) having only its end-vertices in common with the two cycles. By using a new cospectral invariant for ( r , r + 1 ) - almost regular graphs, we will show that almost all dumbbell graphs (without cycle C 4 as a subgraph) are determined by the adjacency spectrum.