Spectral Characterizations of Dumbbell Graphs

Abstract

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.

Graphs determined by signless Laplacian spectra

“…We can obtain DQS graphs with independent edges and isolated vertices from Theorem 3.2. [23,30,45,46]. We can obtain DQS graphs with independent edges and isolated vertices from Theorem 3.3.…”

Towards a spectral theory of graphs based on the signless Laplacian, III

“…Graphs consisting of two cycles with just a vertex in common are called ∞graphs in [49]. It is proved that ∞-graphs without triangles are characterized by their Laplacian spectra and that all ∞-graphs, with one exception, are characterized by their signless Laplacian spectra.…”

“…In Part II we have added the following 11 references: [1], [3], [24], [25], [26], [27], [34], [36], [48], [52], [54]. In the meantime the following 16 papers [2], [13], [18], [19], [21], [28], [29], [30], [31], [32], [38], [42], [45], [49], [51], [53] have been published or are in the process of publication at this moment. Together with [11], [12] and this paper, there are in this moment about 50 papers on the signless Laplacian spectrum published since 2005.…”

Toward a Laplacian spectral determination of signed ∞-graphs

“…Spectral determinations have been widely studied for unsigned graphs and recently this kind of study has been taken to signed graph spectra, as well. The spectral determination of (unsigned) ∞-graphs, which is the coalescence between two cycles C r and C s and related graphs, has been already considered in [16]. The spectral determination of signed lollipop graph, which is the coalescence between a cycle and a path, has been studied and it is shown that a signed lollipop graph is determined by the spectrum of its Laplacian matrix (or simply DLS) [5].…”