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The energy of a simple graph $G$ arising in chemical physics, denoted by $mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if ${G_n}$ is a sequence of finite simple graphs with bounded average degree and ${G_n}$ a sequence of spanning subgraphs of ${G_n}$ such that almost all vertices of $G_n$ and $G_n$ have the same degrees, then $G_n$ and $G_n$ have the same asymptotic energy. Thus, for each type of lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions, we have the same asymptotic energy. As applications, we obtain the asymptotic formulae of energies per vertex of the triangular, $3^3.4^2$, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously.
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In this paper we are interested in the asymptotic enumeration of Cayley graphs. It has previously been shown that almost every Cayley digraph has the smallest possible automorphism group: that is, it is a digraphical regular representation (DRR). In this paper, we approach the corresponding question for undirected Cayley graphs. The situation is complicated by the fact that there are two infinite families of groups that do not admit any graphical regular representation (GRR). The strategy for digraphs involved analysing separately the cases where the regular group $R$ has a nontrivial proper normal subgroup $N$ with the property that the automorphism group of the digraph fixes each $N$-coset setwise, and the cases where it does not. In this paper, we deal with undirected graphs in the case where the regular group has such a nontrivial proper normal subgroup.
Building on previous work by the present authors [Proc. London Math. Soc. 119(2):358--378, 2019], we obtain a precise asymptotic estimate for the number $g_n$ of labelled 4-regular planar graphs. Our estimate is of the form $g_n sim gcdot n^{-7/2} rho^{-n} n!$, where $g>0$ is a constant and $rho approx 0.24377$ is the radius of convergence of the generating function $sum_{nge 0}g_n x^n/n!$, and conforms to the universal pattern obtained previously in the enumeration of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to work with large systems of polynomials equations. In particular, we use evaluation and Lagrange interpolation in order to compute resultants of multivariate polynomials. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.
In 1978 Gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues, and ever since then graph energy has been intensively studied. Since graph energy is the trace norm of the adjacency matrix, matrix norms provide a natural background for its study. Thus, this paper surveys research on matrix norms that aims to expand and advance the study of graph energy. The focus is exclusively on the Ky Fan and the Schatten norms, both generalizing and enriching the trace norm. As it turns out, the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics. The survey brings to the fore the exceptional role of Hadamard matrices, conference matrices, and conference graphs in matrix norms. In addition, a vast new matrix class is studied, a relaxation of symmetric Hadamard matrices. The survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics. Thus, open problems and questions are raised to outline topics for further investigation.
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.
Answering some questions of Gutman, we show that, except for four specific trees, every connected graph G of order n, with no cycle of order 4 and with maximum degree at most 3, has energy greater that its order. Here, the energy of a graph is the sum of the moduli of its eigenvalues. We give more general theorems and state two conjectures.
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