A dissociation set of a graph G is a vertex subset S contained in V(G) such that the induced subgraph G[S] has maximum degree at most 1, and the dissociation number of G is the cardinality of a maximum dissociation set denoted by {\Phi}(G). In this paper, we first derive a sharp lower bound on the dissociation number for all unicyclic graphs. Furthermore, we also study the graphs with minimum spectral radius among all unicyclic graphs with given order and dissociation number and explicitly identify all extremal graphs for \(n-4 \le \Phi(G) \le n-1\).
Let G be a simple connected graph. A set S\subseteq V(G) is called a perfect dominating set if every vertex v\in V(G)\backslash S is adjacent to exactly one vertex in S. The perfect domination number of G, denoted by \gamma_p(G), is the minimum cardinality of such a set. In this paper, we investigate the perfect domina...
Edilberto Cuaresma, Airish Pelemeniano-Jumonong· International Journal of Mat...· 0 citations
An independent set of a graph G is a subset of VG, no two of which are adjacent. The cardinality of a maximum independent set in a graph G is called the independence number of G, denoted by alpha(G). The essential connectivity kappa'(G) of a graph G is denoted as the minimum number of vertices of G whose removal produc...
Shuang-Lv-Ren-Jiang-Hou-Xue-Gong-Li-Ying-Jie-Deng- Ding, Dan Li, Yuan-Yuan Chen· 0 citations
The (Albertson) irregularity of a graph G is irr ( G ) = ∑︁ | d ( u ) − d ( v ) | , where the sum is over all edges of G . In 2005, Hansen and M´elot proved an upper bound on the irregularity of a graph on n vertices and m edges that is attained for all feasible values, and characterized the extremal graphs. We give a...
Allan Bickle· Discrete Mathematics Letters· 0 citations
For fixed $p\in(0,1)$, we prove that the maximum number of vertices in an induced subgraph of $G(n,p)$ from a given class is concentrated on two consecutive integers, unifying and extending several earlier results. We require that the class contain arbitrarily large graphs and that every sufficiently large $k$-vertex m...
Margarita Akhmejanova, V. Kozhevnikov· 0 citations
Let $\tau(G)$ denote the maximum number of pairwise edge-disjoint spanning trees contained in a graph $G$. Zhang and Fan [J. Graph Theory, 2026] posed the problem of determining the corresponding tight spectral-radius condition for every $s$-edge-connected graph with \[ k+1\leq s\leq2k-1, \] together with a characteriz...
Jia Wei, Mu-Huo Liu, Zhi-Fu You et al.· 0 citations
A graph is called a threshold graph if it is induced-$\{P_4,C_4,2K_2\}$ free. In 2020, Qiao and Zhan [J. Graph Theory 93 (2020) 222--229] determined the minimum number of Hamilton cycles in a Hamiltonian threshold graph and characterized the extremal graphs attaining this minimum. We show that every Hamiltonian thresho...
Cheng-Li Li, Feng Liu· Electronic Journal of Combin...· 0 citations
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