Consider the following general question: if we can solve Maximum Matching in (quasi) linear time on a graph class C , does same hold true for of graphs that be modularly decomposed into ? What makes latter question difficult is problem not preserved by quotient, thereby making to exploit structural properties quotient subgraphs modular decomposition. So far, are only aware recent framework (Cou...

Kemeny’s constant of a simple connected graph G is the expected length random walk from i to any given vertex $$j \ne i$$ . We provide method for computing 1-separable graphs via effective resistance methods electrical network theory. Using this formula, we furnish proof that path on n vertices maximizes class undirected trees vertices. Applying again, simplify existing expressions barbell and ...

Given two graphs H1 and H2, a graph is (H1,H2)-free if it contains no induced subgraph isomorphic to or H2. Let Pt be the path on t vertices. A bull obtained from triangle with disjoint pendant edges. In this paper, we show that there are finitely many 5-vertex-critical (P5, bull)-free graphs.

In this paper, we show that in the class of connected graphs G of order n ≥ 3 having girth at least equal to k, 3 ≤ k ≤ n, the unique graph G having minimum general sum-connectivity index χα(G) consists of Ck and n−k pendant vertices adjacent to a unique vertex of Ck, if−1 ≤ α < 0. This property does not hold for zeroth-order general Randić index Rα(G).

Let $$\Gamma =(K_{n},H^-)$$ be a signed complete graph whose negative edges induce subgraph H. The index of $$\Gamma$$ is the largest eigenvalue its adjacency matrix. In this paper, we study when H unicyclic graph. We show that among all graphs order $$n>5$$ k and maximizes index, triangle with remaining vertices being pendant at same vertex triangle.

Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...