arXiv subjects

Ravindra B. Bapat

Publications and source records attributed to Ravindra B. Bapat.

6 recordsLinked to original sources

arXiv2024

Computing the permanental polynomial of $4k$-intercyclic bipartite graphs

Let $G$ be a bipartite graph with adjacency matrix $A(G)$. The characteristic polynomial $ϕ(G,x)=\det(xI-A(G))$ and the permanental polynomial $π(G,x) = \text{per}(xI-A(G))$ are both graph invariants used to distinguish graphs. For bipartite graphs, we define the modified characteristic polynomial, which is obtained by changing the signs of some of the coefficients of $ϕ(G,x)$. For $4k$-intercyclic bipartite graphs, i.e., those for which the removal of any $4k$-cycle results in a $C_{4k}$-free graph, we provide an expression for $π(G,x)$ in terms of the modified characteristic polynomial of the graph and its subgraphs. Our approach is purely combinatorial in contrast to the Pfaffian orientation method found in the literature to compute the permanental polynomial.

math.CO↗

arXiv2019

Resistance distance in directed cactus graphs

Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set $V=\{1,\dotsc,n\}$. The classical distance $d_{ij}$ between any two vertices $i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$ and $j$. The resistance distance (or, simply the resistance) between any two vertices $i$ and $j$ in $V$ is defined by $r_{ij}:=l_{ii}^†+l_{jj}^†-2l_{ij}^†$, where $l_{pq}^{\dagger}$ is the $(p,q)^{\rm th}$ entry of the Moore-Penrose inverse of $L$ which is the Laplacian matrix of $G$. In practice, the resistance $r_{ij}$ is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between $i$ and $j$ is always less than or equal to the classical distance, i.e. $r_{ij} \leq d_{ij}$. However, no proof for this inequality is known. In this paper, we show that this inequality holds for all directed cactus graphs.

math.CO↗

arXiv2018

Squared distance matrix of a weighted tree

Let $T$ be a tree with vertex set $\{1, \ldots, n\}$ such that each edge is assigned a nonzero weight. The squared distance matrix of $T,$ denoted by $Δ,$ is the $n \times n$ matrix with $(i,j)$-element $d(i,j)^2,$ where $d(i,j)$ is the sum of the weights of the edges on the $(ij)$-path. We obtain a formula for the determinant of $Δ.$ A formula for $Δ^{-1}$ is also obtained, under certain conditions. The results generalize known formulas for the unweighted case.

math.CO↗

arXiv2018

Eigenvalues of weakly balanced signed graphs and graphs with negative cliques

In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having negative cliques. This includes cycle graphs, path graphs, complete graphs with vertex-disjoint negative cliques of different orders, and star block graphs with negative cliques. Interestingly, if we reverse the signs of the edges of these graphs, we get the families of weakly balanced signed graphs, thus the eigenvalues of wide classes of weakly balanced signed graphs are also calculated. In social network theory, the eigenvalues of the signed graphs play an important role in determining their stability and developing the measures for the degree of balance.

cs.DM↗

arXiv2016

kirchhoff index and degree kirchhoff index of complete multipartite graph

The Kirchhoff index of a graph is defined as half of the sum of all effective resistance distances between any two vertices. Assuming a complete multipartite graph G, by methods from linear algebra we explicitly formulate effective resistance distances between any two vertices of G, and its Kirchhoff index. In rest of paper we explore extremal value of Kirchhoff index for multipartite graphs

math.CO↗

arXiv2014

Monotonicity properties of certain Laplacian eigenvectors associated with trees

Nath and Paul (Linear Algebra Appl.,460(2014),97-110) have shown that the largest distance Laplacian eigenvalue of a path is simple and the corresponding eigenvector has properties similar to the Fiedler vector. We given an alternative proof, establishing a more general result in the process. It is conjectured that a similar phenomenon holds for any tree.

math.CO↗