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Register a new userSpectra of comb graphs with tails
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Authors
Leonid Golinskii
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Given two graphs, a backbone and a finger, a comb product is a new graph obtained by grafting a copy of the finger into each vertex of the backbone. We study the comb graphs in the case when both components are the paths of order $n$ and $k$, respectively, as well as the above comb graphs with an infinite ray attached to some of their vertices. A detailed spectral analysis is carried out in both situations.
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Two method for computation of the spectra of certain infinite graphs are suggested. The first one can be viewed as a reversed Gram--Schmidt orthogonalization procedure. It relies heavily on the spectral theory of Jacobi matrices. The second method is related to the Schur complement for block matrices. A number of examples including infinite graphs with tails, chains of cycles and ladders are worked out in detail.
We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove a fundamental connection between the symmetry of the spectrum and the existence of damped two-periodic solutions for the discrete-time heat equation on the graph.
In his survey Beyond graph energy: Norms of graphs and matrices (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph, respectively. We show that these graphs have at most two nonzero distinct absolute eigenvalues and investigate the proposed problems organizing our study according to the type of spectrum they can have. In most cases all graphs are characterized. Infinite families of graphs are given otherwise. We also show that all graphs satifying the properties required in the problems are integral, except for complete bipartite graphs $K_{p,q}$ and disconnected graphs with a connected component $K_{p,q}$, where $pq$ is not a perfect square.
Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown to be of special interest, as they are extreme in the sense of an Alon-Boppana theorem, and they have remarkable combinatorial features, such as small diameter, Chernoff bound for sampling, optimal covering time and sharp cutoff. Other topics explored are the connection to Cayley graphs and digraphs, the spectral radius of universal covers, Alons conjecture for random digraphs, and explicit constructions of almost-normal Ramanujan digraphs.
A caterpillar graph $T(p_1, ldots, p_r)$ of order $n= r+sum_{i=1}^r p_i$, $rgeq 2$, is a tree such that removing all its pendent vertices gives rise to a path of order $r$. In this paper we establish a necessary and sufficient condition for a real number to be an eigenvalue of the Randic matrix of $T(p_1, ldots, p_r)$. This result is applied to determine the extremal caterpillars for the Randic energy of $T(p_1,ldots, p_r)$ for cases $r=2$ (the double star) and $r=3$. We characterize the extremal caterpillars for $r=2$. Moreover, we study the family of caterpillars $Tbig(p,n-p-q-3,qbig)$ of order $n$, where $q$ is a function of $p$, and we characterize the extremal caterpillars for three cases: $q=p$, $q=n-p-b-3$ and $q=b$, for $bin {1,ldots,n-6}$ fixed. Some illustrative examples are included.
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