Papers by Hilal A . Ganie
Discrete mathematics algorithms and applications/Discrete mathematics, algorithms, and applications, Mar 7, 2024
Mathematics, Jan 5, 2024
This article is an open access article distributed under the terms and conditions of the Creative... more This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY
Carpathian Mathematical Publications, Dec 29, 2023
Journal of Clinical Interventional Radiology ISVIR, 2019
Blister aneurysms pose significant diagnostic and therapeutic challenge to neurointerventionists ... more Blister aneurysms pose significant diagnostic and therapeutic challenge to neurointerventionists as well as neurosurgeons. Fragile nature of these aneurysms with involvement of the adjacent parent artery makes surgical options more difficult and complicated. Various endovascular treatment options such as overlapping stent, parent artery occlusion, and flow diverter placement are available in the present era. Though side wall aneurysms can be managed with flow diverter placement, bifurcation aneurysms are difficult to treat with these devices due to possibility of compromised flow in covered branch arteries. Blister aneurysms of the middle cerebral artery (MCA) are rare, and their treatment can be challenging when they are treated by endovascular methods with either overlapping stent or flow diverter placement. The authors report a case of ruptured MCA trifurcation blister aneurysm treated with shelfing technique using braided stent monotherapy along with coil embolization. The patie...
Indian Journal of Pure and Applied Mathematics
Applied Mathematics and Computation, Apr 1, 2021
For a simple connected graph G of order n having distance Laplacian eigenvalues ρ L 1 ≥ ρ L 2 ≥ •... more For a simple connected graph G of order n having distance Laplacian eigenvalues ρ L 1 ≥ ρ L 2 ≥ • • • ≥ ρ L n , the distance Laplacian energy DLE(G) is defined as DLE(G) = n i =1 ρ L i − 2 W (G) n , where W (G) is the Weiner index of G. In this paper, we describe the distance Laplacian eigenvalues of a tree of diameter 3. We discuss the distance Laplacian energy of trees of diameter 3 and show that like the Laplacian energy [31] these trees can be ordered on the basis of their distance Laplacian energy. As application we obtain an upper bound for the distance Laplacian energy of a connected graph.
Discrete Mathematics, Algorithms and Applications, Oct 20, 2022
For a simple connected graph [Formula: see text], the generalized distance matrix [Formula: see t... more For a simple connected graph [Formula: see text], the generalized distance matrix [Formula: see text] is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are, respectively, the diagonal matrix of vertex transmission degrees and the distance matrix of [Formula: see text]. In this paper, we will study the spectral radius of the generalized distance matrix [Formula: see text] of a connected graph [Formula: see text]. We obtain some upper and lower bounds for the generalized distance spectral radius in terms of various parameters, like the vertex transmission degrees, the average transmission 2-degrees, etc., associated with the structure of the graph. We characterize the extremal graphs attaining these bounds. Further, we show that our bounds improve some previously known bounds existing in the literature.
Discrete Mathematics, Algorithms and Applications
Let [Formula: see text] be a connected graph with [Formula: see text] vertices and [Formula: see ... more Let [Formula: see text] be a connected graph with [Formula: see text] vertices and [Formula: see text] edges. Let [Formula: see text] be the digraph obtained by orienting the edges of [Formula: see text] arbitrarily. The digraph [Formula: see text] is called an orientation of [Formula: see text] or oriented graph corresponding to [Formula: see text]. The skew Laplacian matrix of the digraph [Formula: see text] is denoted by [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] is the skew matrix and [Formula: see text] is the diagonal matrix with [Formula: see text]th diagonal entry [Formula: see text]. In this paper, we obtain combinatorial representation for the first five coefficients of characteristic polynomial of skew Laplacian matrix of [Formula: see text]. We provide examples of orientations of some well-known graphs to highlight the importance of our results. We conclude the paper with some observations about the skew Laplacian spectral determ...
Applied Mathematics and Computation, 2021
For a simple connected graph G of order n having distance Laplacian eigenvalues ρ L 1 ≥ ρ L 2 ≥ •... more For a simple connected graph G of order n having distance Laplacian eigenvalues ρ L 1 ≥ ρ L 2 ≥ • • • ≥ ρ L n , the distance Laplacian energy DLE(G) is defined as DLE(G) = n i =1 ρ L i − 2 W (G) n , where W (G) is the Weiner index of G. In this paper, we describe the distance Laplacian eigenvalues of a tree of diameter 3. We discuss the distance Laplacian energy of trees of diameter 3 and show that like the Laplacian energy [31] these trees can be ordered on the basis of their distance Laplacian energy. As application we obtain an upper bound for the distance Laplacian energy of a connected graph.
Carpathian Journal of Mathematics, Jul 30, 2022
The normalized distance Laplacian matrix of a connected graph G, denoted by D L (G), is defined b... more The normalized distance Laplacian matrix of a connected graph G, denoted by D L (G), is defined by D L (G) = T r(G) −1/2 D L (G)T r(G) −1/2 , where D(G) is the distance matrix, the D L (G) is the distance Laplacian matrix and T r(G) is the diagonal matrix of vertex transmissions of G. The set of all eigenvalues of D L (G) including their multiplicities is the normalized distance Laplacian spectrum or D L-spectrum of G. In this paper, we find the D L-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the D L-spectrum of the graphs associated with algebraic structures. In particular, we find the D L-spectrum of the power graphs of groups, the D L-spectrum of the commuting graphs of non-abelian groups and the D L-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.
Applied Mathematics and Computation, May 1, 2023
Heliyon, Mar 1, 2022
Let D be a digraph of order n and with a arcs. The signless Laplacian matrix Q(D) of D is defined... more Let D be a digraph of order n and with a arcs. The signless Laplacian matrix Q(D) of D is defined as Q(D)=Deg(D)+A(D), where A(D) is the adjacency matrix and Deg(D) is the diagonal matrix of vertex out-degrees of D. Among the eigenvalues of Q(D) the eigenvalue with largest modulus is the signless Laplacian spectral radius or the Q-spectral radius of D. The main contribution of this paper is a series of new lower bounds for the Q-spectral radius in terms of the number of vertices n, the number of arcs, the vertex out-degrees, the number of closed walks of length 2 of the digraph D. We characterize the extremal digraphs attaining these bounds. Further, as applications we obtain some bounds for the signless Laplacian energy of a digraph D and characterize the extremal digraphs for these bounds.
Bulletin Of The Brazilian Mathematical Society, New Series, Sep 30, 2022
Quaestiones Mathematicae, Feb 3, 2023
Discrete Mathematics, Algorithms and Applications
Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be ... more Let [Formula: see text] be a digraph of order [Formula: see text] and let [Formula: see text] be the adjacency matrix of [Formula: see text] Let Deg[Formula: see text] be the diagonal matrix of vertex out-degrees of [Formula: see text] For any real [Formula: see text] the generalized adjacency matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text] This matrix generalizes the spectral theories of the adjacency matrix and the signless Laplacian matrix of [Formula: see text]. In this paper, we find [Formula: see text]-spectrum of the joined union of digraphs in terms of spectrum of adjacency matrices of its components and the eigenvalues of an auxiliary matrix determined by the joined union. We determine the [Formula: see text]-spectrum of join of two regular digraphs and the join of a regular digraph with the union of two regular digraphs of distinct degrees. As applications, we obtain the [Formula: see text]-spectrum of various families of uns...
arXiv (Cornell University), Jul 22, 2019
For a simple connected graph G, let D(G), T r(G), D L (G) and D Q (G), respectively be the distan... more For a simple connected graph G, let D(G), T r(G), D L (G) and D Q (G), respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of a graph G. The convex linear combinations D α (G) of T r(G) and D(G) is defined as D α (G) = αT r(G) + (1 − α)D(G), 0 ≤ α ≤ 1. As D 0 (G) = D(G), 2D 1 2 (G) = D Q (G), D 1 (G) = T r(G) and D α (G) − D β (G) = (α − β)D L (G), this matrix reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. Let ∂ 1 (G) ≥ ∂ 2 (G) ≥ • • • ≥ ∂ n (G) be the eigenvalues of D α (G) and let D α S(G) = ∂ 1 (G) −∂ n (G) be the generalized distance spectral spread of the graph G. In this paper, we obtain some bounds for the generalized distance spectral spread D α (G). We also obtain relation between the generalized distance spectral spread D α (G) and the distance spectral spread S D (G). Further, we obtain the lower bounds for D α S(G) of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for D α S(G) in terms of clique number and independence number of the graph G and characterize the extremal graphs for some cases.
Ars Mathematica Contemporanea, Oct 5, 2022
For a simple graph G, the generalized adjacency matrix A α (G) is defined as A α (G) = αD(G) + (1... more For a simple graph G, the generalized adjacency matrix A α (G) is defined as A α (G) = αD(G) + (1 − α)A(G), α ∈ [0, 1], where A(G) is the adjacency matrix and D(G) is the diagonal matrix of the vertex degrees. It is clear that A 0 (G) = A(G) and 2A 1 2 (G) = Q(G) implying that the matrix A α (G) is a generalization of the adjacency matrix and the signless Laplacian matrix. In this paper, we obtain some new upper and lower bounds for the generalized adjacency spectral radius λ(A α (G)), in terms of vertex degrees, average vertex 2-degrees, the order, the size, etc. The extremal graphs attaining these bounds are characterized. We will show that our bounds are better than some of the already known bounds for some classes of graphs. We derive a general upper bound for λ(A α (G)), in terms of vertex degrees and positive real numbers b i. As application, we obtain some new upper bounds for λ(A α (G)). Further, we obtain some relations between clique number ω(G), independence number γ(G) and the generalized adjacency eigenvalues of a graph G.
Linear Algebra and its Applications, Nov 1, 2022
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Papers by Hilal A . Ganie