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This paper provides new upper bounds on the spectral radius \ (largest eigenvalue of the adjacency matrix) of graphs embeddable on a given compact surface. Our method is to bound the maximum rowsum in a polynomial of the adjacency matrix, using simple consequences of Euler's formula. Let # denote the Euler genus (the number of crosscaps plus twice the number of handles) of a fixed surface 7. Then (i) for n 3, every n-vertex graph embeddable on 7 has \ 2+-2n+8#&6, and (ii) a 4-connected graph with a spherical or 4-representative embedding on 7 has \ 1+-2n+2#&3. Result (i) is not sharp, as Guiduli and Hayes have recently proved that the maximum value of \ is 3Â2+-2n+o(1) as n Ä for graphs embeddable on a fixed surface. However, (i) is the only known bound that is computable, valid for all n 3, and asymptotic to -2n like the actual maximum value of \. Result (ii) is sharp for the sphere or plane (#=0), with equality holding if and only if the graph is a``double wheel'' 2K 1 +C n&2 (+ denotes join). For other surfaces we show that (ii) is within O(1Ân 1Â2 ) of sharpness. We also show that a recent bound on \ by Hong can be deduced by our method. Academic Press

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Robertson conjectured that the only 3-connected, internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph.  We provide a counterexample to this conjecture.

Robertson has conjectured that the only 3-connected internally 4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord is the Petersen graph. We prove this conjecture in the special case where the graphs involved are also cubic. Moreover, this proof does not require the internal-4-connectivity assumption. An example is then presented to show that the assumption of internal 4-connectivity cannot be dropped as an hypothesis in the original conjecture. We then summarize our results aimed toward the solution of the conjecture in its original form. In particular, let $G$ be any 3-connected internally-4-connected graph of girth 5 in which every odd cycle of length greater than 5 has a chord. If $C$ is any girth cycle in $G$ then $N(C)\backslash V(C)$ cannot be edgeless, and if $N(C) \backslash V(C)$ contains a path of length at least 2, then the conjecture is true. Consequently, if the conjecture is false and $H$ is a counterexample, then for any girth cycle $C$ in $H$, $N(C) \backslash V(C)$ induces a nontrivial matching $M$ together with an independent set of vertices. Moreover, $M$ can be partitioned into (at most) two disjoint non-empty sets where we can precisely describe how these sets are attached to cycle $C$.