On the nullity of cycle-spliced $\mathbb{T}$-gain graphs

Ciampella, Adriana; Khan, Suliman

doi:10.22049/cco.2024.29772.2155

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	On the nullity of cycle-spliced $\mathbb{T}$-gain graphs
Communications in Combinatorics and Optimization
مقاله 9، دوره 10، شماره 2، شهریور 2025، صفحه 381-403 اصل مقاله (537.95 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2024.29772.2155
نویسندگان
Adriana Ciampella^* ¹؛ Suliman Khan²
¹Department of Mathematics and Applications, University of Naples “Federico II”, Piazzale Tecchio 80, Napoli 80125, Italy
²Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, Viale Lincoln 5, Caserta, I–81100, Italy
چکیده
Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain (or complex unit gain) graph and $A(\Phi)$ be its adjacency matrix. The nullity of $\Phi$, denoted by $\eta(\Phi)$, is the multiplicity of zero as an eigenvalue of $A(\Phi)$, and the cyclomatic number of $\Phi$ is defined by $c(\Phi)=e(\Phi)-n(\Phi)+\kappa(\Phi)$, where $n(\Phi)$, $e(\Phi)$ and $\kappa(\Phi)$ are the number of vertices, edges and connected components of $\Phi$, respectively. A connected graph is said to be cycle-spliced if every block in it is a cycle. We consider the nullity of cycle-spliced $\mathbb{T}$-gain graphs. Given a cycle-spliced $\mathbb{T}$-gain graph $\Phi$ with $c(\Phi)$ cycles, we prove that $0 \leq \eta(\Phi)\leq c(\Phi)+1$. Moreover, we show that there is no cycle-spliced $\mathbb{T}$-gain graph $\Phi$ of any order with $\eta(\Phi)=c(\Phi)$ whenever there are no odd cycles whose gain has real part $0$. We give examples of cycle-spliced $\mathbb{T}$-gain graphs whose nullity equals the cyclomatic number, and we show some properties of those graphs $\Phi$ such that $\eta(\Phi)=c(\Phi)-\varepsilon$, $\varepsilon \in \{0,1\}$. A characterization is given in case $\eta(\Phi)=c(\Phi)$ when $\Phi$ is obtained by identifying a unique common vertex of $2$ cycle-spliced $\mathbb{T}$-gain graphs $\Phi_1$ and $\Phi_2$. Finally, we compute the nullity of all $\mathbb{T}$-gain graphs $\Phi$ with $c(\Phi)=2$.
کلیدواژه‌ها
Cyclomatic number؛ Zero eigenvalue multiplicity؛ Complex unit gain graphs

مراجع
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On the nullity of cycle-spliced $\mathbb{T}$-gain graphs