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The well-known Moore bound $M(k,g)$ serves as a universal lower bound for the order of $k$-regular graphs of girth $g$. The excess $e$ of a $k$-regular graph $G$ of girth $g$ and order $n$ is the difference between its order $n$ and the corresponding Moore bound, $e=n - M(k,g) $. We find infinite families of parameters $(k,g)$, $g$ even, for which we show that the excess of any $k$-regular graph of girth $g$ is larger than $4$. This yields new improved lower bounds on the order of $k$-regular graphs of girth $g$ of smallest possible order; the so-called $(k,g)$-cages. We also show that the excess of the smallest $k$-regular graphs of girth $g$ can be arbitrarily large for a restricted family of $(k,g)$-graphs satisfying a very natural additional structural property.

The Moore bound M (k, g) is a lower bound on the order of k-regular graphs of girth g (denoted (k, g)-graphs). The excess e of a (k, g)-graph of order n is the difference n − M (k, g). In this paper we consider the existence of (k, g)bipartite graphs of excess 4 via studying spectral properties of their adjacency matrices. We prove that the (k, g)-bipartite graphs of excess 4 satisfy thewhere A denotes the adjacency matrix of the graph in question, J the n × n all-ones matrix, E the adjacency matrix of a union of vertex-disjoint cycles, and H d−1 (x) is the Dickson polynomial of the second kind with parameter k − 1 and of degree d − 1. We observe that the eigenvalues other than ±k of these graphs are roots of the polynomials H d−1 (x) + λ, where λ is an eigenvalue of E. Based on the irreducibility of H d−1 (x) ± 2 we give necessary conditions for the existence of these graphs. If E is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic excess; if E is the adjacency matrix of a disjoint union of two cycles we call the corresponding graphs graphs with bicyclic excess. In this paper we prove the non-existence of (k, g)-graphs with cyclic excess 4 if k ≥ 6 and k ≡ 1 (mod 3), g = 8, 12, 16 or k ≡ 2 (mod 3), g = 8, and the non-existence of (k, g)-graphs with bicyclic excess 4 if k ≥ 7 is odd number and g = 2d such that d ≥ 4 is even.

Let G be a graph on n vertices and m edges, with maximum degree Δ(G) and minimum degree δ(G). Let A be the adjacency matrix of G, and let λ1≥λ2≥…≥λn be the eigenvalues of G. The energy of G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G, that is E(G)=|λ1|+…+|λn|. The energy of G is known to be at least twice the minimum degree of G, E(G)≥2δ(G). Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G, i.e., E(G)≥Δ(G)+δ(G). In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E(G). The results rely on elementary inequalities and their application.

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