The weighted vertex PI index of a graph G is defined by PIw(G)=?e=uv?E(G) (dG(u)+dG(v))(nu(e|G) + nv(e|G)) where dG(u) denotes the vertex degree of u and nu(e|G) denotes the number of vertices in G whose distance to the vertex u is smaller than the distance to the vertex v. A graph is a cactus if it is connected and all its blocks are either edges or cycles. In this paper, we give the upper and lower bounds on the weighted vertex PI index of cacti with n vertices and s cycles, and completely characterize the corresponding extremal graphs.

In this paper, we consider (a,b)- parity factors in graphs and obtain a toughness condition for the existence of (a,b)-parity factors. Furthermore, we show that the result is sharp in some sense.

Regular graphs are classical symmetric structures in graph theory, where each vertex has identical degree and the overall topology often exhibits strong automorphism properties. However, practical systems frequently require heterogeneous constraints, which can be modeled by introducing vertex colorings and non-uniform degree requirements, leading to controlled symmetry breaking. In this paper, we investigate two-tone factors in edge-connected regular graphs and claw-free cubic graphs under arbitrary red-blue vertex colorings. Using the framework of parity (g,f)-factors, we establish two main existence results. First, we prove that every λ-edge connected r-regular graph admits a two-tone ({k},{k,k+2})-factor for any coloring, provided that r/λ⩽k⩽r−r/λ, k⩽r−2, and k|G| is even. Second, we show that every 3-edge connected claw-free cubic graph admits a two-tone ({0,1},{2,3})-factor regardless of the coloring configuration. Beyond existence, we provide a constructive algorithm by reducing the parity factor problem to an exact f-factor problem and further to a perfect matching problem via vertex-splitting techniques. We rigorously justify the correctness of this reduction and show that the desired factor can be computed in polynomial time. From a structural perspective, our results reveal that edge-connectivity serves as a stabilizing mechanism that preserves parity feasibility under arbitrary color-induced perturbations, while claw-free constraints enforce local density that prevents parity imbalance. This provides a symmetry-based interpretation of two-tone factors as a balance between global regularity and local asymmetry. These findings contribute to both the theoretical development of factor theory and its algorithmic realization, with potential implications for deterministic network design and resource allocation in structured systems.

We study two-tone factors in edge-connected regular graphs and claw-free cubic graphs under red-blue vertex colorings. A two-tone factor is a spanning subgraph in which vertices of different colors are assigned different allowed degree sets. For edge-connected regular graphs, we prove an existence theorem for two-tone $\left(\right\{k\},\{k,k+2\left\}\right)$-factors under arbitrary red-blue colorings and describe a constructive factor-theoretic route to matching-type formulations. For the claw-free cubic setting, we show that the natural arbitrary-coloring extension for $\left(\right\{0,1\},\{2,3\left\}\right)$-factors is false in general. Nevertheless, we prove a restricted positive result when the graph has a triangle decomposition and the coloring is constant on each triangle, and we give a mask-consistency characterization for triangle-inflated cubic graphs. The computational section is therefore framed as an implementation and proof-audit study: it certifies recovered factors in the regular case, cross-checks claw-free cubic stress tests by independent MILP and mask-CSP formulations, and explains why only provable restrictions are retained as theorems. From the perspective of symmetry, the contrast identifies a boundary between symmetry-stable and symmetry-fragile colored degree constraints.