Let SPEX_𝒫(n,F) and SPEX_𝒪𝒫(n,F) denote the sets of graphs with the maximum spectral radius over all n-vertex F-free planar and outerplanar graphs, respectively. Define tP_ℓ as a linear forest of t vertex-disjoint ℓ -paths and P_t ·ℓ as a starlike tree with t branches each of length ℓ - 1 . Building on the structural framework by Tait and Tobin [25] and the work of Fang, Lin and Shi [12] on the planar spectral extremal graphs without vertex-disjoint cycles, this paper determines the extremal graphs in SPEX_𝒫(n,tP_ℓ ) and SPEX_𝒪𝒫(n,tP_ℓ ) for sufficiently large n. When t=1 , since tP_ℓ is a path of a specific length, our results adapt Nikiforov’s findings [22] under the (outer)planarity condition. When ℓ =2 , tP_ℓ corresponds to a matching of size t, and our work generalizes the results of Wang, Huang and Lin [26] as well as Yin and Li [Discrete Math., 2026]. Motivated by the results of Zhai and Liu [31], we further solve the edge-disjoint paths version and determine the extremal graphs in SPEX_𝒫(n,P_t ·ℓ) and SPEX_𝒪𝒫(n,P_t ·ℓ) for sufficiently large n.
The decycling number ∇ (G) (connected decycling number ∇ _1(G) ) of a graph G is the smallest number of vertices which can be deleted from G so that the resulting graph is a forest (a subtree). A decycling set with ∇ (G) vertices is said to be a ∇ -set of G. The Betti deficiency ξ (G) of G is the minimum number of odd components among all cotrees of G. In this article, we investigate the structure and distribution of ∇ -sets in cubic graphs by topological techniques. We show that for any ∇ -set S of a cubic graph G, there is a Xuong-tree T_X of G such that S consists of the leaves of T_X with at most ξ (G) exceptions, where ξ (G) is the Betti deficiency of G. Moreover, we present a sufficient and necessary condition for ∇ _1(G)=∇ (G) for a cubic graph G.
Let p be bounded away from zero and one, let r∼ 2log _1/pn , and suppose that the expected number μ _r of copies of K_r in G(n, p) is of order n^1+θ . Morris and Riordan recently obtained a precise upper and lower point-probability description of the clique hypergraph when -1<θ <1/2 . We show that their upper point-probability estimate extends to the natural range -1<θ <1 . The error changes at θ =1/2 : if Δ _k denotes the expected number of connected k-clusters of copies of K_r , then, for every fixed k≥ 3 , Δ _k=O^* (n^θ +n^3θ -1 ), and this order is sharp for k=3 . The local contribution is exactly ( [ r+1; 3 ]) μ _r+1 ; the second term comes from dispersed clusters. This yields a transition in polynomial order at θ =1/2 . We also prove a general variance inequality for cluster counts and deduce that the three-cluster count is concentrated throughout 0<θ <1 .
An injective edge-coloring of a graph G is an assignment of colors to the edges of G such that two distinct edges e and f receive distinct colors whenever 1) e and f are in a common triangle; or 2) {e,f} forms a matching and there is a third edge adjacent to both e and f. The minimum number of colors needed for an injective edge-coloring of a graph G is called the injective chromatic index of G, denoted by χ '_inj(G) . In this paper, we prove the following:
A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. A brick is either solid or nonsolid. For a matching covered graph G, let (G) denote the rank of its perfect matching lattice. For a brick G, (G)=|E(G)|-|V(G)|+1 , and therefore G has at least (G) perfect matchings. We call G extremal if it has exactly (G) perfect matchings and sub-extremal if it has exactly (G)+1 perfect matchings. Carvalho, Lucchesi and Murty established structural results on extremal nonsolid bricks. In this paper, we present analogous structural results on sub-extremal nonsolid bricks.
